to number the various categories, objects, arrows, and 2-cells we need to introduce and discuss.
while I think about which, if any, should be the primary numbering system.
This is a somewhat technical, nitty-gritty, topic, involving some fine details,
but it plays a crucial role in proofs of adjoint functor theorems (see Chapter X, Section 1, of Mac Lane's CWM; also its Chapter V, Section 6, both the Lemma stated there and the exercises to that section), and is worth a close look, where details get filled in, to see what is entailed.
$\calK$
Much of this post is an experiment in presentation, experimenting with different ways of presenting the proof,
in particular different diagrams that present different aspects, parts, or views of what is going on.
<h3>Comma categories: review and notation</h3>
The notation in the proof gets a bit sticky.
There are both notational and conceptual issues.
Let us introduce this topic by comparing
(the universal 1-cone $\rightadj{ \langle \pi_{\rightcat 1}, \pi_{\rightcat 2} \rangle }$ of a very familiar categorical product, $\R\times\R \cong \R^2$), to
(the universal 2-diagrams ($\rightcat Q [\leftcat\lambda]$) for $(l \downarrow \calL)$ and $(l \downarrow \rightadj R)$, the comma categories in question).
(For the 1-universal property of those, see Exercise II.6.4 of CWM.
For a characterization of those as weighted limits (cylinders) see Kelly 1989 [EO2CL] (4.2) .)
The diagram at left below, for $\R\times\R \cong \R^2$, is in the 1-category of real vector spaces $\Vect$,
the other diagrams are 2-diagrams in the 2-category $\CAT$.
{Further diagrams are below that top diagram.)
\[ \leftcat{ \boxed{ \begin{array} {ccccc|ccccc|ccccc|ccccc} & \leftcat{ \boxed{ \R\times\R \cong \R^2 } \rlap{ \lower2.8ex{ \kern-3.2em \boxed{10} } } } & \rightcat{ \xrightarrow[\kern3.5em\boxed{11}\kern3.55em]{\textstyle \pi_2} } & \R & \hskip.5em & \hskip.5em & \leftcat{ \boxed{ (l \downarrow \calL) } \rlap{ \lower3ex{ \kern-2.2em \boxed{10} } } } & \rightcat{ \xrightarrow[\kern2em\boxed{11} \; \text{faithful} \kern2em]{\textstyle Q = r} } & \calL & \hskip.5em & \hskip.5em & \leftcat{ \boxed{ (l \downarrow \rightadj R) } \rlap{ \lower3ex{ \kern-2.2em \boxed{10} } } } & \rightcat{ \xrightarrow[\kern1.5em\boxed{11} \; \text{faithful} \kern1.5em]{\textstyle Q = r} } & \rightcat\calR & \hskip.5em & \hskip.5em & \boxed{ ( l \downarrow {\rightadj R}) } & \xrightarrow[\kern4em]{\textstyle \text{fiber}} & \leftcat{ \boxed{ (\Delta \Downarrow {\rightadj R}) } } & \rightcat{ \xrightarrow[\kern4em]{\textstyle !} } & \calI \\ & \llap{\pi_1} \Bigg\downarrow \rlap{\boxed{11}} & & & \hskip0em & \hskip0em & \llap ! {\Bigg\downarrow} \rlap{\boxed{11}} & \llap{\boxed{11}\;\lambda} \Bigg\Uparrow & \Bigg\Vert \rlap{ 1_\calL \; \boxed{0} } & \hskip1em & \hskip0em & \llap ! {\Bigg\downarrow} \rlap{\boxed{11}} & \llap{\boxed{11}\;\lambda} \Bigg\Uparrow & \Bigg\downarrow \rlap{ \rightadj R \; \boxed{0} } & \hskip1em & \hskip0em & \llap{!} {\Bigg\downarrow} & \rightadj{\text{p.b.}} & \leftcat{\Bigg\downarrow} & \Bigg\Uparrow & \rightadj{ \Bigg\downarrow \rlap{ \ulcorner R \urcorner } }& \\ & \R & & & \hskip0em & \hskip0em & \calI & \xrightarrow[\textstyle \boxed{1} \; l]{} & \calL & \hskip0em & \hskip0em & \calI & \xrightarrow[\textstyle \boxed{1} \; l]{} & \calL & \hskip0em & \hskip0em & \calI & \xrightarrow[\textstyle l]{} & \leftcat\calL & \xrightarrow[\textstyle \Delta]{} & [\rightcat\calR,\leftcat\calL] \\ \end{array} } } \]
\[ \leftcat{ \boxed{ \begin{array} {cc|c|ccc|ccccc|c|ccc|ccc|cccc} \R^2 \rlap{ \lower3.2ex{ \kern-1.4em \boxed{10} } } & \kern0em && \kern0em & (l\downarrow\calL) \rlap{ \lower3.2ex{ \kern-2.3em \boxed{10} } } \rlap{ \kern.5em \rightcat{ \xrightarrow[ \textstyle \kern4em \boxed{11} \; \text{faithful} \kern4em ]{\textstyle Q=r} } } & & & & & \calL \rlap{ \lower3.2ex{ \kern-1em \boxed{0} } } & \kern1em && \kern0em & \calJ \rlap{ \xrightarrow[\kern4.7em \boxed{12} \kern4.7em] {\textstyle \boxed{ \black G = \black G \rightcat Q [\black G \lambda]} } } & \kern0em & \kern0em & (l\downarrow\calL) \rlap{ \lower3.2ex{ \kern-2.3em \boxed{10} } } \rlap{ \rightcat{ \xrightarrow[ \textstyle \kern3.8em \boxed{11} \; \text{faithful} \kern3.8em ]{\textstyle Q=r} } } & \kern0em & & & & \calL \rlap{ \lower3.2ex{ \kern-1em \boxed{0} } } & \\ \\ \hline X = \langle X\pi_1,X\pi_2 \rangle = \langle x,y \rangle &&&& \black X = \black X \rightcat Q [\black X \lambda] = k[\kappa] &&&&& X\rightcat Q = k &&&& j &&& j\black G = j \big( \black G \rightcat Q [\black G \lambda] \big) = j \black G \rightcat Q [j \black G \lambda] &&&&& j\black G\rightcat Q \\ && && &&& & \llap{X\lambda = \kappa} \nearrow & &&&&& &&&&&& \llap{j \black G \lambda} \nearrow & \\ && && \Bigg\downarrow \rlap\mu & & \kern2em & l & \mu & \Bigg\downarrow \rlap{\mu\rightcat Q = \mu} &&&& \llap\iota \Bigg\downarrow &&& \llap{\iota \black G = {} } \Bigg\downarrow \rlap{ {} = \iota \black G \rightcat Q } & && l & \iota \black G & \Bigg\downarrow \rlap{\iota \black G \rightcat Q} & \\ & &&&&&& & \llap{X'\lambda = \kappa'} \searrow & &&&& &&&&& && \llap{j' \black G \lambda} \searrow & \\ &&&& X' = X' \rightcat Q [X' \lambda] = k' [\kappa'] &&&&& X'\rightcat Q = k' &&&& j' &&& j' \black G = j' \big( \black G \rightcat Q [\black G \lambda] \big) = j' \black G \rightcat Q [j' \black G \lambda] &&&&& j' \black G\rightcat Q \\ \end{array} } } \]
\[ \leftcat{ \boxed{ \begin{array} {c|cc|c|cccc|c|cccccc} \calJ \rlap{ \xrightarrow[\textstyle \kern2em \boxed{12} \; \text{right lax fiber} \kern2em]{\textstyle \boxed{ \black G = \black G \rightcat Q [\black G \lambda]} } } & \boxed{ (l\downarrow {\rightadj R}) } \rlap{ \lower3.2ex{ \kern-2.5em \boxed{10} } } \rlap{ \rightcat{ \kern.5em \xrightarrow[ \textstyle \kern5.5em \boxed{11} \; \text{faithful} \kern5.5em ]{\textstyle Q=r} } } & & \rightcat\calR \rlap{ \rightadj{ \kern.5em \xrightarrow[\kern8em \boxed{0} \kern8em]{\textstyle R} } } & & \calL && \kern2em &&&& \CAT \\ \hline j & X = j\black G = j \big( \black G \rightcat Q [\black G \lambda] \big) = j \black G \rightcat Q [j \black G \lambda] = \black X\rightcat Q [\black X \lambda] = \rightcat r [\kappa] && \kern2em \black{X} \rightcat Q = j\black G\rightcat Q = \rightcat r \kern1em & \kern7em && \black{X} \rightcat Q \rightadj R = j\black G\rightcat Q\rightadj R = \rightcat r \rightadj R = k \\ & &&&& \llap{\black{X} \lambda = j \black G \lambda = \kappa} \nearrow & \\ \llap\iota \Bigg\downarrow & \llap{\iota \black G = {}} \Bigg\downarrow \rlap{ {} = \iota \black G \rightcat Q = \rightcat\rho } && \llap{ \iota \black G \rightcat Q \; } \Bigg\downarrow \rlap{ {} = \rightcat\rho} & \kern1em l & {} \rlap{\kern0em \iota \black G = \rightcat\rho} & \Bigg\downarrow \rlap{\iota \black G \rightcat Q\rightadj R = \rightcat\rho \rightadj R = \mu} & & & & \boxed{ (l\downarrow {\rightadj R}) } & \rightcat{ \kern.5em \xrightarrow[ \textstyle \boxed{11} \; \text{faithful} ]{\textstyle Q=r} } & \rightcat\calR \\ && && & \llap{\black{X'} \lambda = j' \black G \lambda = \kappa'} \searrow & \\ j' & X' = j' \black G = j' \big( \black G \rightcat Q [\black G \lambda] \big) = j' \black G \rightcat Q [j' \black G \lambda] = \black{X'} \rightcat Q [\black{X'} \lambda] = \rightcat{r'} [\kappa'] && \black{X'} \rightcat Q = j' \black G\rightcat Q = \rightcat{r'} &&& \black{X'} \rightcat Q \rightadj R = j' \black G\rightcat Q\rightadj R = \rightcat{r'} \rightadj R = k' \\ \hline & \rightadj{ \Bigg\downarrow \rlap{ \kern1em {{\rightcat r} [\kappa]} \mathrel{\rightadj\mapsto} {{(\rightcat r \rightadj R)} [\kappa]} } } &&&&&& &&& \Bigg\downarrow & \rightadj{ \text{p.b.} } & \rightadj{ \Bigg\downarrow\rlap{R \; \boxed{0}} } & \kern1em \\ \hline \calJ \rlap{ \xrightarrow[\textstyle \kern2em \boxed{12} \; \text{right lax fiber} \kern2em]{\textstyle \boxed{ \black G = \black G \rightcat Q [\black G \lambda]} } } & \boxed{ (l\downarrow\calL) } \rlap{ \lower3.2ex{ \kern-2.5em \boxed{10} } } \rlap{ \rightcat{ \xrightarrow[ \textstyle \kern13em \boxed{11} \; \text{faithful} \kern13em ]{\textstyle Q=r} } } & & & &\calL && \\ \hline j & X = j\black G = j \big( \black G \rightcat Q [\black G \lambda] \big) = j \black G \rightcat Q [j \black G \lambda] = \black X\rightcat Q [\black X \lambda] = k[\kappa] &&&&& \black{X} \rightcat Q = j\black G\rightcat Q = k \\ & &&&& \llap{\black{X} \lambda = j \black G \lambda = \kappa} \nearrow & & & \\ \llap\iota \Bigg\downarrow & \llap{\iota \black G = {} } \Bigg\downarrow \rlap{ {} = \iota \black G \rightcat Q = \mu } & && l & {} \rlap{\kern0em \iota \black G = \mu} & \Bigg\downarrow \rlap{\iota \black G \rightcat Q = \mu} & & & & \boxed{ (l\downarrow\calL) } & \rightcat{ \xrightarrow[ \textstyle \boxed{11} \; \text{faithful} ]{\textstyle Q=r} } & \leftcat\calL \\ && && & \llap{\black{X'} \lambda = j' \black G \lambda = \kappa'} \searrow & &&&& \llap{\boxed{11} \; !} \downarrow & \llap{ \boxed{11} \; \lambda } \Uparrow & \Vert \rlap{1_\calL \; \boxed{0}} \\ j' & X' = j' \black G = j' \big( \black G \rightcat Q [\black G \lambda] \big) = j' \black G \rightcat Q [j' \black G \lambda] = \black{X'} \rightcat Q [\black{X'} \lambda] = k'[\kappa'] &&&&& \black{X'} \rightcat Q = j' \black G\rightcat Q = k' &&&& \calI & \xrightarrow[\textstyle \boxed{1} \; l]{} & \calL \\ \end{array} } } \]
\[ \leftcat{ \boxed{ \begin{array} {cccccc|ccccccc|cccccc} &&&& \rightcat y &&&&& \\ 1 & \xrightarrow[\kern3em]{\textstyle \boxed{ X = \langle X\pi_1,X\rightcat{\pi_2}\rangle = \langle x,\rightcat y \rangle } } & \leftcat{ {\R}^2 \rlap{ \lower2.8ex{ \kern-1.3em \boxed{10} } } } & \rightcat{ \xrightarrow[\kern3.5em\boxed{11}\kern3.55em]{\textstyle \pi_2} } & \R & \hskip1em & \hskip1em & & & & & \\ && \llap{\pi_1} \Bigg\downarrow \rlap{\boxed{11}} & & & \hskip1em & \hskip1em & & & & & & \kern1em \\ & x & \R & & & \hskip1em & \hskip1em & &&&& \\ \hline && 1 && \rightadj{\boxed{3}} & \hskip1em & \hskip1em & && 1 && \rightadj{\boxed{3}} & \\ & \llap{!} \nearrow & \red{ \Bigg\downarrow \rlap{ \kern-5em \boxed{\textstyle \boxed{5}\;\big((X\pi_2)\lim_{\leftcat\R}\big)[\overline{X\lambda}]} } } & \rightadj{ \searrow \rlap{(\black X \rightcat{\pi_2})\lim_{\leftcat\R}} } & & \hskip1em & \hskip1em & & \llap{!} \nearrow & \red{ \Bigg\downarrow \rlap{ \kern-5em \boxed{\textstyle \boxed{5}\;\big((X\pi_2)\lim_{\leftcat\R}\big)[\overline{X\lambda}]} } } & \rightadj{ \searrow \rlap{(\black X \rightcat \pi_2)\lim_{\leftcat\R}} } & & \\ &&&& \rightcat y &&&&& \\ (\leftcat a, \rightcat b ) & \xrightarrow[\kern3em]{\textstyle \boxed{ X = \langle X\pi_1,X\rightcat{\pi_2}\rangle = \langle x,\rightcat y \rangle } } & \leftcat{ {\mathbf R}^2 \rlap{ \lower2.8ex{ \kern-1.3em \boxed{10} } } } & \rightcat{ \xrightarrow[\kern3.5em\boxed{11}\kern3.55em]{\textstyle \pi_2} } & \mathbf R & \hskip1em & \hskip1em & \calJ & \xrightarrow[\kern3.5em\boxed{1}\kern3.5em]{\smash{\textstyle G}} & \leftcat{ (l \downarrow \calL) \rlap{ \lower2.8ex{ \kern-2em \boxed{10} } } } & \rightcat{ \xrightarrow[\kern3.5em\boxed{11}\kern3.5em]{\smash{\textstyle Q = r} } } & \calL \\ && \llap{\pi_1} \Bigg\downarrow \rlap{\boxed{11}} & & & \hskip1em & \hskip1em & \Bigg\downarrow & \rightadj{ {} \rlap{ \kern4em \raise3.6ex{\llap{ \boxed{21}\;\pi_{\black G \rightcat Q} } \bigg\Uparrow} } } & \rightadj{ \nearrow \rlap{ \kern-6.2em \lower1ex{ \boxed{20}\;(\black G\rightcat Q)\leftcat{\lim_\calL} } } } & {} \rlap{ \kern-5.4em \lower4ex{ \bigg\Uparrow \rlap{ \overline{\black G\lambda}\;\boxed{32} } } } & \Bigg\Vert \rlap{ 1_\calL \; \boxed{0} } & \kern1em \\ & x & \mathbf R & & & \hskip1em & \hskip1em & 1 \rlap{ \kern1em \xrightarrow[\textstyle \boxed{1} \; l]{\kern21em} } &&&& \calL \\ \hline \\ && 1 && \rightadj{\boxed{20}} & \hskip1em & \hskip1em & && 1 && \rightadj{\boxed{20}} & \\ & \llap{!} \nearrow & \red{ \Bigg\downarrow \rlap{ \kern-5em \boxed{\textstyle \boxed{22}\;\big((GQ)\lim_{\leftcat\calL}\big)[\overline{G\lambda}]} } } & \rightadj{ \searrow \rlap{(\black G \rightcat Q)\lim_{\leftcat\calL}} } & \smash{ \lower3ex {\rightcat k} } & \hskip1em & \hskip1em & & \llap{!} \nearrow & \red{ \Bigg\downarrow \rlap{ \kern-5em \boxed{\textstyle \boxed{22}\;\big((GQ)\lim_{\leftcat\calL}\big)[\overline{G\lambda}]} } } & \rightadj{ \searrow \rlap{(\black G \rightcat Q)\lim_{\leftcat\calL}} } & & \\ \calJ & \xrightarrow [\kern3.5em\boxed{12}\kern3.5em]{\smash{\textstyle \boxed { G = G\rightcat Q [G\lambda] = \rightcat k[\kappa] }}} & \leftcat{ (l \downarrow \calL) \rlap{ \lower2.8ex{ \kern-2em \boxed{10} } } } & \rightcat{ \xrightarrow[\kern3.5em\boxed{11}\kern3.5em]{\textstyle Q = r} } & \calL & \hskip1em & \hskip1em & \calJ & \xrightarrow[\kern3.5em\boxed{12}\kern3.5em]{\smash{\textstyle G}} & \leftcat{ (l \downarrow \calL) \rlap{ \lower2.8ex{ \kern-2em \boxed{10} } } } & \rightcat{ \xrightarrow[\kern3.5em\boxed{11}\kern3.5em]{\smash{\textstyle Q = r} } } & \calL \\ & \text{CWM Exercise V.1.1} & \llap{!} \Bigg\downarrow \rlap{\boxed{11}} & \llap{\boxed{11}\;\lambda} \Bigg\Uparrow & \Bigg\Vert \rlap{ 1_\calL \; \boxed{0} } & \hskip1em & \hskip1em & \Bigg\downarrow & \rightadj{ {} \rlap{ \kern4em \raise3.6ex{ \llap{ \boxed{21} \; \pi_{\black G\rightcat Q} } \bigg\Uparrow } } } & \rightadj{ \nearrow \rlap{\kern-6.2em \lower1ex{\boxed{20}\;(\black G\rightcat Q)\lim_{\leftcat\calL}}} } & {} \rlap{ \kern-5.4em \lower4ex{ \bigg\Uparrow \rlap{ \overline{\black G\lambda}\;\boxed{32} } } } & \Bigg\Vert \rlap{ 1_\calL \; \boxed{0} } & \kern1em \\ && 1 & \xrightarrow[\textstyle \boxed{1} \; l]{} & \calL & \hskip1em & \hskip1em & 1 \rlap{ \kern1em \xrightarrow[\textstyle \boxed{1} \; l]{\kern21em} } &&&& \calL \\ \hline \\ \hline &&&& \rightcat k &&&&& \\ \calI & \xrightarrow [\kern3.5em\boxed{1}\kern3.5em]{\smash{\textstyle \boxed { X = X\rightcat Q [X\lambda] = \rightcat k[\kappa] }}} & \leftcat{ (l \downarrow \calL) \rlap{ \lower2.8ex{ \kern-2em \boxed{0} } } } & \rightcat{ \xrightarrow [\kern3.5em\boxed{0}\kern3.55em]{\textstyle Q = r} } & \calL & \hskip1em & \hskip1em & \calJ & \xrightarrow[\kern3.5em\boxed{1}\kern3.5em]{\smash{\textstyle G}} & \leftcat{ (l \downarrow \calL) \rlap{ \lower2.8ex{ \kern-2em \boxed{0} } } } & \rightcat{ \xrightarrow[\kern3.5em\boxed{0}\kern3.5em]{\smash{\textstyle Q = r} } } & \calL \\ & & \llap{!} \Bigg\downarrow \rlap{\boxed{0}} & \llap{\boxed{0}\;\lambda} \Bigg\Uparrow & \Bigg\Vert \rlap{ 1_\calL \; \boxed{0} } & \hskip1em & \hskip1em & \Bigg\downarrow & \rightadj{ {} \rlap{ \kern4em \raise3.6ex{\llap{\boxed{3} \; \pi_{\black G\rightcat Q}} \bigg\Uparrow} } } & \rightadj{ \nearrow \rlap{\kern-6.2em \lower1ex{\boxed{3}\;(\black G\rightcat Q)\lim}} } & {} \rlap{ \kern-5.4em \lower4ex{ \bigg\Uparrow \rlap{ \overline{\black G\lambda}\;\boxed{4} } } } & \Bigg\Vert \rlap{ 1_\calL \; \boxed{0} } & \kern1em \\ && 1 & \xrightarrow[\textstyle \boxed{0} \; l]{} & \calL & \hskip1em & \hskip1em & 1 \rlap{ \kern1em \xrightarrow[\textstyle \boxed{0} \; l]{\kern21em} } &&&& \calL \\ \hline \\ && 1 && \rightcat{\boxed{20}} & \hskip1em & \hskip1em & && 1 && \rightcat{\boxed{20}} & \\ & \llap{!} \nearrow & \red{ \Bigg\downarrow \rlap{ \kern-5em \boxed{\textstyle \boxed{22} \; \rightcat{ \big( (\leftcat G Q)\lim_\calR \big) } \leftcat{ [\overline{G\lambda}] } } } } & \rightcat{ \searrow \rlap{ (\leftcat G Q)\lim_\calR } } & \smash{ \lower3ex {\rightcat s} } & \hskip1em & \hskip1em & & \llap{!} \nearrow & \red{ \Bigg\downarrow \rlap{ \kern-5em \boxed{\textstyle \boxed{22} \; \rightcat{ \big( (\leftcat G Q)\lim_\calR \big) } \leftcat{ [\overline{G\lambda}] } } } } & \rightcat{ \searrow \rlap{ (\leftcat G Q)\lim_\calR } } & & \\ \calJ & \xrightarrow[\kern3.5em\boxed{12}\kern3.5em]{\smash{\textstyle \boxed { G = G\rightcat Q \leftcat{[G\lambda]} = \rightcat s \leftcat{[\kappa]} }}} & \leftcat{ (l \downarrow \rightadj R) \rlap{ \lower2.8ex{ \kern-2em \boxed{10} } } } & \rightcat{ \xrightarrow[\kern3.5em\boxed{11}\kern3.55em]{\textstyle Q = r} } & \rightcat\calR & \hskip1em & \hskip1em & \calJ & \xrightarrow[\kern3.5em\boxed{12}\kern3.5em]{\smash{\textstyle G}} & \leftcat{ (l \downarrow \rightadj R) \rlap{ \lower2.8ex{ \kern-2em \boxed{10} } } } & \rightcat{ \xrightarrow [\kern3.5em\boxed{11}\kern3.5em]{\smash{\textstyle Q = r} } } & \rightcat\calR \\ & \text{CWM Lemma V.6} & \llap{!} \Bigg\downarrow \rlap{\boxed{11}} & \llap{\boxed{11}\;\lambda} \Bigg\Uparrow & \rightadj{ \Bigg\downarrow \rlap{ R \; \boxed{0} } } & \hskip1em & \hskip1em & \Bigg\downarrow & \rightcat{ {} \rlap{ \kern4em \raise3.6ex{\llap{\boxed{21} \; \pi_{\black G\rightcat Q} } \bigg\Uparrow} } } & \leftadj{ \nearrow \rlap{ \kern-6.2em \lower1ex{ \boxed{20}\;(\black G\rightcat Q)\lim_{\rightcat\calR} } } } & \leftadj{ {} \rlap{ \kern-5.4em \lower4ex{ \bigg\Uparrow \rlap{ \leftcat{ \overline{\black G\leftcat\lambda} } \; \boxed{32} } } } } & \rightadj{ \Bigg\downarrow \rlap{ R \; \boxed{0} } } & \kern1em \\ && 1 & \xrightarrow[\textstyle \boxed{1} \; l]{} & \calL & \hskip1em & \hskip1em & 1 \rlap{ \kern1em \xrightarrow[\textstyle \boxed{1} \; l]{\kern21em} } &&&& \calL \\ \end{array} } } \]
To understand those categories,
a necessary starting point is to see what the objects and arrows in each look like
in relation to
the objects and arrows in their constituent categories.
To that end, consider the above diagrams:
<hr />
(Repitious repeat, to be edited :-)
Before wading into (the existence proof) for (a limit) in (a comma category),
it is useful to review the definitions of (the objects and arrows in comma categories),
and (their relations) to (those in the related categories).
(This is material in Section II.6, Comma Categories, of CWM, adopted with notation tailored to
(a putative right adjoint functor $\rightadj{ \boxed{0} \; R : \rightcat\calR \to \leftcat\calL }$).)
To that end, consider the above (rather large display).
This display accomplishes several things.
First, it shows (objects and arrows in several different categories and how they relate to each other).
Second, it shows how (objects in comma categories) can be broken into (their components) ($X \mapsto X\rightcat Q, X\leftcat\lambda$),
and conversely how (compatible entities) can be combined into (an object in a comma category) ($k, \kappa \mapsto k[\kappa]$),
and, moreover, this not just for (individual objects), but also for (functors into comma categories) (variable objects).
<hr />
Suppose we have a path, a function of time, in $\R^2$
Such a path could be denoted as a single variable, say a single letter like $X$,
or in terms of its coordinates, say $\langle x_1,x_2 \rangle$, or $\langle x,y \rangle$,
or we could introduce explicit coordinate functions on ${\mathbf R}^2$, say $\pi_1$ and $\pi_2$
and view the coordinates as composites $X\pi_1$ and $X\pi_2$,
which can be combined to give $X$ by: $\boxed{ X = \langle X\pi_1,X\pi_2 \rangle }$.
(I.e., do we show the coordinates of $X$, or merely $X$ as a single entity?)
The same choices arise for functors into comma categories.
($\leftcat{ (l\downarrow\calL) }$) has its "coordinate functions"
($\rightcat Q : \leftcat{ (l\downarrow\calL) } \to \calL$) and ($ \leftcat{ \lambda : {!l} \Rightarrow \rightcat Q : {(l\downarrow\calL)} \to \calL } $).
Thus below I speak about (a functor $G : \calJ \to {(l\downarrow\calL)} $), and its "coordinates" $G\rightcat Q$ and $G\leftcat\lambda$,
which can be put back together to get $G$ by: $\boxed{ G = G\rightcat Q \leftcat{ [\black G\lambda] } }$,
which acts as follows on ($\iota : j \to j'$ in $\calJ$):
\[ \leftcat{ \boxed{ \begin{array} {cc|ccc|cccc} \calJ \rlap{ \xrightarrow[\kern11em]{\textstyle \boxed{ \black G = \black G \rightcat Q [\black G \lambda]} } } & \kern0em & \kern0em & (l\downarrow\calL) \rlap{ \rightcat{ \xrightarrow[ \textstyle \kern4.5em \text{faithful} \kern4.5em ]{\textstyle Q=r} } } & & & & &\calL &9 \\ \\ \hline j &&& j\black G = j \big( \black G \rightcat Q [\black G \lambda] \big) = j \black G \rightcat Q [j \black G \lambda] &&&&& j\black G\rightcat Q \\ &&& &&&& \llap{j \black G \lambda} \nearrow & \\ \llap\iota \Bigg\downarrow &&& \llap{\iota \black G = {} } \Bigg\downarrow \rlap{ {} = \iota \black G \rightcat Q } & && l & \iota \black G & \Bigg\downarrow \rlap{\iota \black G \rightcat Q} & \\ &&&& && & \llap{j' \black G \lambda} \searrow & \\ j' &&& j' \black G = j' \big( \black G \rightcat Q [\black G \lambda] \big) = j' \black G \rightcat Q [j' \black G \lambda] &&&&& j' \black G\rightcat Q \\ \end{array} } } \]
<hr />
In the simplest situation (CWM Exercise V.1.1),
we start with (a single category $\leftcat{ \boxed{0} \; \calL }$) and (an object $\leftcat{ \boxed{1} \; l \in \calL }$).
We can then form (the comma category $\leftcat{ \boxed{\boxed{10}\;(l\downarrow \calL)} }$)
with its universal 2-diagram in the 2-category $\CAT$,
one leg of which is the (canonical faithful, but not full, forgetful functor
$\rightcat{ \boxed{\leftcat{(l\downarrow \calL)} \xrightarrow[\kern1em\boxed{11}\kern1em]{\textstyle Q = r} \leftcat\calL : \leftcat{k[\kappa : l \to k] \mathrel{\rightcat\mapsto} k}} }$).
(The boxed numbers are used in some diagrams below.)
Suppose now we have
(a functor $\boxed{ \boxed{12} \; G : \calJ \to \leftcat{(l\downarrow \calL)} }$) such that
(the composite $\boxed{ 12; \rightcat{11} } \; \black G\rightcat Q$) has
(a limit cone $\rightadj{ \boxed{\textstyle \boxed{21}\; \pi_{\black G \rightcat Q} : \boxed{20} \; (\black G\rightcat Q)\lim_{\leftcat\calL} \Rightarrow \black G\rightcat Q} }$ in $\leftcat\calL$).
\[ \leftcat{ \boxed{ \begin{array} {cccccc|ccccccc|cccccc} && 1 && \rightadj{\boxed{20}} & \hskip1em & \hskip1em & && 1 && \rightadj{\boxed{20}} & \\ & \llap{!} \nearrow & \red{ \Bigg\downarrow \rlap{ \kern-5em \boxed{\textstyle \boxed{22} \; \big((GQ)\lim_{\leftcat\calL}\big)[\overline{G\lambda}]} } } & \rightadj{ \searrow \rlap{(\black G \rightcat Q)\lim_{\leftcat\calL}} } & & \hskip1em & \hskip1em & & \llap{!} \nearrow & \red{ \Bigg\downarrow \rlap{ \kern-5em \boxed{\textstyle \boxed{22}\;\big((GQ)\lim_{\leftcat\calL}\big)[\overline{G\lambda}]} } } & \rightadj{ \searrow \rlap{(\black G \rightcat Q)\lim_{\leftcat\calL}} } & & \\ \calJ & \xrightarrow [\kern3.5em \boxed{12} \kern3.5em] {\smash{\textstyle G}} & \leftcat{ (l \downarrow \calL) \rlap{ \lower2.8ex{ \kern-2em \boxed{10} } } } & \rightcat{ \xrightarrow [\kern3.5em\boxed{11}\kern3.55em]{\textstyle Q = r} } & \calL & \hskip1em & \hskip1em & \calJ & \xrightarrow[\kern3.5em\boxed{12}\kern3.5em]{\smash{\textstyle G}} & \leftcat{ (l \downarrow \calL) \rlap{ \lower2.8ex{ \kern-2em \boxed{10} } } } & \rightcat{ \xrightarrow[\kern3.5em\boxed{11}\kern3.5em]{\smash{\textstyle Q = r} } } & \calL \\ {} \rlap{\kern-2em \text{CWM Exercise V.1.1}} && \Bigg\downarrow & \llap{\boxed{11}\;\lambda} \Bigg\Uparrow & \Bigg\Vert \rlap{ 1_\calL \; \boxed{0} } & \hskip1em & \hskip1em & \Bigg\downarrow & \rightadj{ {} \rlap{ \kern4em \raise3.6ex{\llap{ \boxed{21} \; \pi_{\black G \rightcat Q} } \bigg\Uparrow} } } & \rightadj{ \nearrow \rlap{ \kern-6.2em \lower1ex{ \boxed{20}\;(\black G\rightcat Q)\lim_{\leftcat\calL} } } } & {} \rlap{ \kern-5.4em \lower4ex{ \bigg\Uparrow \rlap{ \overline{\black G\lambda}\;\boxed{32} } } } & \Bigg\Vert \rlap{ 1_\calL \; \boxed{0} } & \kern1em \\ && 1 & \xrightarrow[\textstyle \boxed{1} \; l]{} & \calL & \hskip1em & \hskip1em & 1 \rlap{ \kern1em \xrightarrow[\textstyle \boxed{1} \; l]{\kern25em} } &&&& \calL \\ \end{array} } } \]
See (the external 2-diagrams in $\CAT$ above), also (the internal diagrams in two functor categories) in (the first row below).
Ignore for the moment (the second row).
\[ \leftcat{ \boxed{ \begin{array} {cccc l|ccccccccc} && [ \calJ, \leftcat{(l\downarrow \calL)} ] {} \rlap{ \kern.5em \rightcat{ \xrightarrow[\kern37em]{\textstyle [ \leftcat\calJ, (Q=r) ] } } } &&& && &&& [ \calJ, \calL ] && \\ \hline && \llap{\boxed{1 | 12}\;} G && & \kern1em & \kern1em && \llap{ \boxed{2 | 12;\rightcat{11}} \kern1em } G\rightcat Q && \\ && & \rightadj{ \nwarrow \rlap{ \pi_{\black G} \; \boxed{5 | 121} } } & && & \llap{G\lambda} \nearrow & \smash{ \lower1ex{ \boxed{ \scriptstyle \text{defn } \overline{G\lambda} } } } & \rightadj{ \nwarrow \rlap{ \pi_{\black G \rightcat Q} \; \boxed{3 | 21} } } \\ && && \rightadj{\big((\black G\rightcat Q)\lim_{\leftcat\calL}\big)}[ \overline{G\lambda} ] \; \boxed{5 | 132} && \leftcat{\llap{\boxed{0}\kern1em} l} & {} \rlap{\kern-1em \xrightarrow[ \textstyle \boxed{ \exists!\overline{G\lambda} \; \boxed{4 | 32} } ]{\kern12em} } &&& \rightadj{ (\black G\rightcat Q)\lim_{\leftcat\calL} \; \boxed{3 | 20} } \\ && {} \rlap{ \kern3em \text{a single $\calJ$-cone in $(l\downarrow \calL)$} } &&&& {} \rlap{\kern0em \text{mediating arrow, a morphism of $\calJ$-cones in $\calL$} } &&&& \\ \hline \forall \; \boxed{6} \; k[\kappa] & {} \rlap{ \kern-1em \xrightarrow[\kern17em]{\textstyle \forall \mu \; \boxed{7}} } &&& \black G \rlap{\; \boxed{1}} \kern1em & \kern1em & & \boxed{6\rightcat Q} & k & {} \rlap{ \kern-1em \xrightarrow [\kern12em] {\textstyle \mu\rightcat Q = \mu \; \boxed{7}} } &&& \black G\rightcat Q & \boxed{2} \\ & \llap{ \exists! \; \boxed{8'} \; \bar\mu} \searrow && \rightadj{ \nearrow \rlap{ \pi_{\black G} \; \boxed{5 | 121} } } & & && \llap{\kappa} \nearrow && \llap{\black G\lambda} \nearrow \rlap{\boxed{2 | 12;11}} && \rightadj{ \nearrow \rlap{ \pi_{\black G \rightcat Q \; \boxed{3 | 21}} } } & \\ && \rightadj{ \big((\black G\rightcat Q)\lim_{\leftcat\calL}\big) } [\overline{\black G\lambda}] & \rightadj{\boxed{5 | 120}} & && \llap{\boxed{0}\kern1em} l & {} \rlap{ \kern-1em \xrightarrow [\textstyle \exists!\overline{\black G\lambda} \; \boxed{4 | 32}] {\smash{\kern12em}} } &&& \rightadj{(\black G\rightcat Q)\lim_{\leftcat\calL}} & \rightadj{\boxed{3 | 20}} & \\ \end{array} } } \]
Then ($\boxed{2}\; G\leftcat\lambda$ is a cone over $\boxed{2}\; G\rightcat Q$),
($\rightadj{ \boxed{3} \; \pi_{\black G \black Q} }$ is a universal cone over $\boxed{2}\; G\rightcat Q$),
so there is a unique (mediating arrow $\leftcat{ \boxed{ \boxed{4} \; \overline{{\black G}\lambda} } }$) such that $\leftcat{ \boxed{ { {\overline{\black G\lambda}} \cdot { \rightadj{ \pi_{\black G \rightcat Q} } } = {\black G\lambda} } } }$.
Note that ($\rightadj{ \boxed{5}\; \big((\black G\rightcat Q)\lim_{\leftcat\calL}\big)\leftcat{[\overline{\black G\lambda}]} }$ is an object in the comma category $\leftcat{\boxed{0}\; (l\downarrow \calL)}$), and
**** (the universal cone $\rightadj{\boxed{3} \; \pi_{\black G \rightcat Q} : (\black G\rightcat Q)\lim_{\leftcat\calL} \Rightarrow \black G\rightcat Q}$ in $\leftcat\calL$)
may also be viewed as
(a cone $\rightadj{ \boxed{\textstyle \boxed{5}\; \pi_{\black G} : \big((\black G\rightcat Q)\lim_{\leftcat\calL}\big)[\overline{\black G\lambda}] \Rightarrow G} }$ in $\leftcat{(l\downarrow \calL)}$) ****.
To show ($\rightcat{\boxed{0}\; Q}$ created a limit) we must show
(the above lift ${\rightadj\pi}_{\rightcat G}$) is (unique as a lift) and (universal as a cone).
For (uniqueness as a lift)
suppose (a cone $\rho : k[\kappa] \Rightarrow G$ in $\leftcat{(l\downarrow \calL)}$)
is (a lift through $\rightcat Q$) of (the cone $\rightadj{\pi_{\black G \rightcat Q} : (\black G\rightcat Q)\lim_{\leftcat\calL} \Rightarrow \black G\rightcat Q}$ in $\calL$).
\[ \leftcat{ \boxed{ \begin{array} {cccc|cccccc} {} \rlap{ \kern2em (l \downarrow \calL) \kern.5em \rightcat{ \xrightarrow[\kern24em]{\textstyle Q = r} } } &&&& & && \calL \\ \hline G &&& \kern1em & \kern1em & && G\rightcat Q && \\ & \nwarrow \rlap\rho && \text{a single cone in } \leftcat{(l\downarrow \calL)} && & \llap{G\lambda} \nearrow && \nwarrow \rlap{ \rho\rightcat Q = \rightadj{ \pi_{\black G \rightcat Q} } } & \\ && k[\kappa] &&& \leftcat l & {} \rlap{\kern-1em \xrightarrow[\textstyle \kappa]{\kern9em} } &&& (k[\kappa])\rightcat Q = \rightadj{ (\black G\rightcat Q)\lim_{\leftcat\calL} } \\ &&&&& {} \rlap{\kern0em \text{a morphism of cones in } \leftcat\calL} &&&& \\ \end{array} } } \]
Then $k \xlongequal{ \text{ defn. of } \rightcat Q \, } (k[\kappa])\rightcat Q \xlongequal{ \text{ lift } } \rightadj{ (\black G\rightcat Q)\lim_{\leftcat\calL} }$
and $\rho = \rho\rightcat Q = \rightadj\pi_{\black G \rightcat Q}$
so $\Big( \rho : k[\kappa] \Rightarrow G \Big)$ equals $\rightadj{ \Big( \pi_{\black G} : \big((\black G\rightcat Q)\lim_{\leftcat\calL}\big)[\kappa] \Rightarrow G \Big) }$ as (cones in $\leftcat{(l\downarrow \calL)}$).
The only variable yet to be determined is $\kappa$.
But, like $\overline{G\lambda}$,
($\kappa$ is a morphism of cones from $G\lambda$ to $\pi_{\black G \rightcat Q}$),
and (the uniqueness clause of the universal property of the latter) gives
$\kappa = \overline{G\lambda}$.
Our job now is to show
(the cone $\rightadj{ \boxed{ \textstyle \boxed{5} \; \pi_{\black G} : \leftcat{ \big((\black G\rightcat Q)\lim_{\leftcat\calL}\big)\Big[\overline{\black G\lambda}\Big] } \Rightarrow \black G } }$)
is (a universal cone to $G$ in $\leftcat{(l\downarrow \calL)}$), thus
$\red{ \boxed{ \textstyle \leftcat{ \big((\black G\rightcat Q) \lim_{\leftcat\calL}\big) \Big[\overline{\black G\lambda}\Big] } = \black G\lim_{ \leftcat{(l\downarrow \calL)} } } }$.
(Consult the internal diagrams below;
numbers in boxes refer to entities in the categories $\leftcat\calL$ and $\leftcat{(l\downarrow \calL)}$.)
\[ \leftcat{ \boxed{ \begin{array} {ccc|c rcccccl} \kern1em && \kern-10em & \kern1em &&& && \llap{ \boxed{1 | 12} \; } \black G && \\ & \leftcat{ \llap{ \boxed{10} \; } (l\downarrow \calL) } &&& && && & \rightadj{ \nwarrow \rlap{\pi_{\black G} \; \boxed{5 | 121}} } & \kern4em \text{(a single cone) in } \leftcat{(l\downarrow \calL)} \\ & \smash{ \lower2ex{ \rightcat{ \llap{ \boxed{11} \; Q } \Bigg\downarrow } } } &&& &&& && & \rightadj{ \big((\black G\rightcat Q) \lim_{\leftcat\calL}\big) } \Big[\overline{\black G\lambda}\Big] \rightadj{ {} = \black G \lim_{ \leftcat{(l\downarrow \calL)} } \; \boxed{5 | 120} } \\ \hline &&&& && && \llap{ \boxed{2 | 12;11} \; } G\rightcat Q && \\ &&&& && & \leftcat{ \llap{\boxed{2 | 12;11 , 31} \; G\lambda} \nearrow } && \rightadj{ \nwarrow \rlap{\pi_{\black G \black Q} \; \boxed{3 | 21}} } & \kern4em \rightadj{ \text{(a single, universal, cone) in } \leftcat\calL } \\ & \leftcat{ \llap{ \boxed{0} \; } \calL } &&& && \leftcat{ \boxed{0 | 1 , 30} \; l } & {} \rlap{\kern-1em \xrightarrow[\textstyle \llap{\boxed{4 | 32} \;} \exists!\overline{G\lambda}] {\kern19em} } &&& \rightadj{ (\black G\rightcat Q)\lim_{\leftcat\calL} \rlap{ \; \boxed{3 | 20} } } \\ &&&&& {} \rlap{ \kern0em \text{definition of $\leftcat{\overline{\black G\lambda}}$ makes it (a morphism of cones in $\leftcat\calL$) : $\leftcat{ \overline{\black G\lambda} : \black G \lambda \to \rightadj{ \pi_{\black G \rightcat Q} } }$ } } &&&&& \\ \hline \\ \hline &&&&& {} \rlap{ \text{($\mu : k[\kappa] \Rightarrow G$ an arbitrary cone) in } \leftcat{(l\downarrow \calL)} \text{ to the base } G } \\ &&&& & & \forall \; \boxed{6} \; k[\kappa] & {} \rlap{ \kern-1em \xrightarrow[\kern18em] {\textstyle \forall\mu \kern1em \boxed{7}} } &&& \black G \rlap{\; \boxed{1 | 12}} \\ & \leftcat{ \llap{ \boxed{10} \; } (l\downarrow \calL) } &&& && & \llap{\boxed{8'} \kern1em \exists!\bar\mu} \searrow && \rightadj{ \nearrow \rlap{ \pi_{\black G} \; \boxed{5 | 121} } } & \kern4em \text{(a single cone) in } \leftcat{(l\downarrow \calL)} \\ & \smash{ \lower2ex{ \rightcat{ \llap{ \boxed{11} \; Q} \Bigg\downarrow } } } &&& && && \rightadj{ \big((\black G\rightcat Q)\lim_{\leftcat\calL}\big) } \Big[\overline{\black G\lambda}\Big] \rlap{ \; \rightadj{ \boxed{5 | 120} } } && \\ \hline &&& {} \rlap{ \text{(the $\rightcat Q$-image of (the above cone $\mu : k[\kappa] \Rightarrow G$)) in $(l\downarrow \leftcat\calL)$ is (a cone $\mu\rightcat Q = \mu : k \Rightarrow G\rightcat Q$) in } \leftcat\calL, } \\ &&& {} \rlap{ \text{thus there exists a unique mediating arrow $\boxed{ \textstyle \boxed{8}\; \bar\mu : k \to \rightadj{ (\black G\rightcat Q)\lim_{\leftcat\calL} } }$ (not shown) in $\calL$} } \\ &&& {} \rlap{ \text{such that $\boxed{8}\; \bar\mu : \mu\rightcat Q \to {\rightadj\pi}_{\black G\rightcat Q}$ is a morphism of cones in $\leftcat\calL$} } \\ &&&& & \boxed{6 \rightcat Q} & k & {} \rlap{ \kern-2em \xrightarrow [\kern20em] {\textstyle \mu \rightcat Q = \mu \kern1em \boxed{7 \rightcat Q}} } &&& \black G\rightcat Q \rlap{ \; \boxed{ 2 | 12;\rightcat{11} } } \\ & \llap{ \boxed{0} \; } \calL &&& & \llap{\kappa} \nearrow \rlap{ \kern-.5em { \text{$\leftcat\mu$ is} \atop \text{a cone in $\leftcat{(l\downarrow\calL)}$} } } && {} \rlap{ \kern0em \black G\lambda \nearrow \kern0em \boxed{2} \kern0em { \text{$\overline{\black G\leftcat\lambda}$ is an arrow} \atop \text{between cones in } \calL } } && \rightadj{ \nearrow \rlap{ \pi_{\black G \rightcat Q} \; \boxed{3 | 21} } } & \kern4em \rightadj{ \text{(a single, universal, cone) in } \leftcat\calL } \\ &&& \kern0em & \llap{ \boxed{0} \; } l & {} \rlap{ \kern-1em \xrightarrow [\textstyle \exists!\overline{\black G\lambda} \rlap{\; \boxed{4 | 32}}] {\kern15em} } &&& \rightadj{ (\black G\rightcat Q)\lim_{\leftcat\calL} \rlap{ \; \boxed{3 | 20} } } && \\ \end{array} } } \]
To that end, let ($\leftcat{\boxed{6} \; k[\kappa]}$ be an object in $\leftcat{(l\downarrow \calL)}$),
and ($\leftcat{\boxed{7}\; \mu : k[\kappa] \Rightarrow G}$ be a cone in $\leftcat{(l\downarrow \calL)}$).
Then ($\boxed{7 \rightcat Q}\; \mu \rightcat Q = \leftcat{\mu : k \Rightarrow \black G\rightcat Q}$ is a cone in $\leftcat\calL$),
so there is a unique (mediating arrow $\leftcat{ \boxed{ \textstyle \boxed{8}\; \bar\mu : k \to \rightadj{ (\black G\rightcat Q)\lim_{\leftcat\calL} } } }$)
(not shown in the diagram) such that ($\leftcat{ \bar\mu : \mu \to \rightadj\pi }$ is a morphism of cones in $\leftcat\calL$).
It remains to show ($\leftcat{ \bar\mu : k \to \rightadj{(\black G\rightcat Q)\lim_{\leftcat\calL}} }$ in $\leftcat\calL$)
is (an arrow $\leftcat{ \bar\mu : k[\kappa] \to \rightadj{\big((\black G\rightcat Q)\lim_{\leftcat\calL}\big)} \leftcat{\Big[\overline{\black G\lambda}\Big]} }$ in $\leftcat{(l\downarrow \calL)}$).
Since $\leftcat{ \kappa\bar\mu \rightadj\pi = \kappa\mu = \black G\lambda = \overline{\black G\lambda}\rightadj\pi }$,
the uniqueness clause in the universal property of $\rightadj\pi$ gives $\leftcat{ \kappa\bar\mu = \overline{\black G\lambda}}$,
so $\leftcat{ \bar\mu : \mu \to \rightadj\pi}$ is an arrow in $\leftcat{ (l\downarrow \calL) }$
and $\leftcat{ \boxed{8'} \; \bar\mu : \mu \to \rightadj\pi }$ is a morphism of cones in $\leftcat{(l\downarrow \calL)}$ (see the diagram).
Further $\leftcat{ \bar\mu : k[\kappa] \to \rightadj{\big((\black G\rightcat Q)\lim_{\leftcat\calL}\big)}\leftcat{[\overline{\black G\lambda}]} }$ in $\leftcat{(l\downarrow \calL)}$
is the unique arrow in $\leftcat{(l\downarrow \calL)}$ such that $\leftcat{\bar\mu} \rightadj\pi = \rightcat\mu$.
Thus $\rightadj{ \pi : \big((\black G \rightcat Q)\lim_{\leftcat\calL}\big) \leftcat{ \big[ \overline{\black G \lambda} \big) ] } \Rightarrow \black G }$ is a universal cone in $\leftcat{(l\downarrow \calL)}$.
In those circumstances $\leftcat{ G : \calJ \to (l\downarrow \calL) }$
has a limit $\rightadj{\pi : \big((\black G \rightcat Q)\lim_{\leftcat\calL}\big)\leftcat{ [ \overline{\black G \lambda} ] } \Rightarrow \black G}$ in $\leftcat{(l\downarrow \calL)}$
which is preserved by the canonical forgetful functor $\leftcat{(l\downarrow \calL) \to \calL}$.
The above diagrams were internal to respectively $\leftcat\calL$ and $\leftcat{(l\downarrow \calL)}$.
There is an external diagram (in $\CAT$) that is useful in showing relations between the limits:
\[ \leftcat{ \begin{array} {} && 1 \\ & \llap{!} \nearrow & \red{ \Bigg\downarrow \rlap{ \kern-4em \boxed{\textstyle \big((GQ)\lim_{\leftcat\calL}\big) \big[\bar\lambda\big]} } } & \searrow \rlap{ (G\rightcat Q)\lim_{\leftcat\calL} } \\ \calJ & \xrightarrow[\textstyle G]{\kern4em} & \boxed{(l \downarrow \calL)} & \rightcat{ \xrightarrow[\textstyle Q = r]{\kern4em} } & \calL \\ \big\Vert &&&& \big\Vert \\ \calJ & {} \rlap{ \kern-2em \xrightarrow[\textstyle G\rightcat Q]{\kern17em} } &&& \calL\end{array} } \]
<hr />
For the adjoint functor theorems we need a two-category, or "relative," version of the above.
(The boxed numbers below are used in both the text and the diagrams.
They give a sense of the order in which various constructions are made.)
Thus we start with a functor $\rightadj{ \boxed{ {\leftcat\calL} \leftarrow {\rightcat\calR} : R \; \boxed{0} } }$
from ($\rightcat{ \boxed{0} \calR} $, the "right category") (that is my terminology; I do not know how common that terminology is)
to ($\leftcat{ \boxed{0} \calL }$, the "left category").
Given $\rightadj R$ and (an object $\leftcat{ \boxed{0|1} \; l \in \calL }$ in the left category $\leftcat\calL$),
we consider (the comma category $\leftcat{ \boxed{ \boxed{0|10} \; (l \downarrow {\rightadj R})} }$)
and its relations (in the 2-category $\CAT$) to several other entities:
\[ \leftcat{ \boxed{ \begin{array} {cccccc|cc} && 1 & \rightcat{ \boxed{3 | 20} } & & \kern2em & \kern2em & && 1 & \rightcat{ \boxed{3 | 20} } & \kern2em \\ & \llap{!} \nearrow & \red{ \Bigg\downarrow \rlap{ \kern-2.5em \boxed{ \textstyle \leftcat G\lim_{ \leftcat{(l \downarrow {\rightadj R})} } } } } & \rightcat{ \searrow \rlap{(\leftcat G Q)\lim_\calR} } & & & & & \llap{!} \nearrow & \red{ \Bigg\downarrow \rlap{ \kern-2.5em \boxed{ \textstyle \leftcat G\lim_{ \leftcat{(l \downarrow {\rightadj R})} } } } } & \rightcat{ \searrow \rlap{(\leftcat G Q)\lim_\calR} } & \\ \calJ & \xrightarrow[\kern1em \boxed{1 | 12} \kern1em]{\textstyle G} & \boxed{(l \downarrow {\rightadj R})} \rlap{ \lower3ex{ \kern-2.6em \boxed{0 | 10} } } & \rightcat{ \xrightarrow[\kern1em \boxed{0 | 11} \kern1em]{\textstyle Q = r} } & \rightcat\calR & & & \calJ & \xrightarrow[\kern1em \boxed{1 | 12} \kern1em]{\textstyle G} & \boxed{ (l \downarrow {\rightadj R}) } \rlap{ \lower3ex{ \kern-2.6em \boxed{0 | 10} } } & \rightcat{ \xrightarrow[\kern1em \boxed{0 | 11} \kern1em]{\textstyle Q = r} } & \rightcat\calR & \\ && \Bigg\downarrow & \llap{ \boxed{0 | 11} \; \lambda } \Bigg\Uparrow & \rightadj{\Bigg\downarrow \rlap{ R \; \boxed{0} } } & {} \rlap{\kern1em \cong} & & {} \rlap{ \kern-2em { \textstyle\text{CWM} \atop \textstyle\text{Exercise V.6.3} } } && \Bigg\downarrow & \rightadj{ \llap{ \boxed{0 | 10} \; } \text{p.b.} } & \rightadj{\Bigg\downarrow \rlap{ R \; \boxed{0} } } \\ && 1 & \xrightarrow[\smash{\textstyle \boxed{0 | 1} \; l}]{} & \calL & & & && \leftcat{(l \downarrow \calL)} & \rightcat{ \xrightarrow{} } & \calL \\ {} \rlap{\kern1em \text{CWM Lemma V.6, Exercise V.6.1} } &&&&&&& {} \rlap{ \kern-2em { \textstyle\text{CWM} \atop \textstyle\text{Exercise V.1.1} } } && \Bigg\downarrow & \llap{ \boxed{0 | 11} \; \lambda } \Bigg\Uparrow & \Bigg\Vert \rlap{ 1_\calL \; \boxed{0} } \\ &&&&&&&&& 1 & \xrightarrow[\textstyle \boxed{0 | 1} \; l]{} & \calL \\ \end{array} } } \]
Whereas the diagram above was (an external diagram, in $\CAT$),
the diagrams below are (internal to the three categories).
\[ \leftcat{ \boxed{ \begin{array} {cccccc c|cccccc c|cccccc} &&& (l\downarrow \rightadj R) \; \boxed{10} \rlap{ \kern.5em \rightcat{ \xrightarrow[\kern28em]{\textstyle Q = r \; \boxed{11}} } } &&&&& && \rightcat{ \calR \; \boxed{0} } \rlap{ \kern.5em \rightadj{ \xrightarrow[\kern35em]{\textstyle R \; \boxed{0}} } } &&&&&&&&& \calL \; \boxed{0} \\ \hline & \forall \; \boxed{6 | 130} \; \rightcat r \leftcat{[\kappa]} & \rightcat{ {} \rlap{ \kern-1em \xrightarrow[ \textstyle \kern3em \text{a cone in } \leftcat{(l\downarrow \rightadj R)} \kern3em ]{\textstyle \forall \rho \; \boxed{7 | 131}} } } &&& G\;\boxed{1 | 12} & \kern1em & \kern1em & \boxed{6 \rightcat Q | 30}\;\rightcat{r} & \rightcat{ {} \rlap{ \kern-1em \xrightarrow[ \textstyle \kern2.5em \text{a cone in } \calR \kern2.5em ]{ \textstyle \rho Q = \rho\;\boxed{7Q | 31} } } } &&& G \rightcat{ Q \; \boxed{2} } & \kern1em & \kern1em &&& \llap{ \boxed{6 \rightcat Q \rightadj R | 30 \rightadj R} \; } {\rightcat r}{\rightadj R} & \rightcat{ {} \rlap{ \kern-3em \xrightarrow [\kern19em] {\textstyle \rho Q \rightadj R = \rho \rightadj R \; \boxed{7 \rightcat Q \rightadj R | 31\rightadj R} } } } &&& G \rightcat{Q} {\rightadj R} \; \boxed{2\rightadj R} & 8 \\ && \rightcat{ \llap{\exists! \; \boxed{8' | 132} \; \bar\rho} \searrow } && \rightcat{ \nearrow \rlap{ \pi_{\black G} \; \boxed{5 | 121} } } & &&& & \rightcat{ \llap{\exists! \; \boxed{8 | 32} \; \bar\rho} \searrow } && \rightcat{ \nearrow \rlap{ \pi_{\black G\rightcat Q} \; \boxed{3 | 21} } } &&&& & \llap{ \boxed{6\lambda} \; \kappa } \nearrow & \kern-1em { \text{$\rightcat\rho$ is a} \atop \text{cone in $\leftcat{ (l\downarrow{\rightadj R}) }$} } \kern1em & \llap{\black G\lambda} \nearrow \rlap{ \boxed{2} } & { \text{$\overline{\black G\leftcat\lambda}$ is an arrow} \atop \text{between cones in $\leftcat\calL$} } & \rightadj{ \nearrow \rlap{ \pi_{\black G\rightcat Q} R \; \boxed{\rightcat 3 R | 21 R} } } && \kern1em \rightadj{ \text{(a single, universal, cone) in } \leftcat\calL } \\ &&& \rightcat{\big((\black G Q)\lim_\calR\big)}[\overline{\black G\lambda}] \rlap{ \; \rightadj{ \boxed{5 | 120} } } && && &&& \rightcat{(\black G Q)\lim_\calR} \rlap{ \rightcat{\; \boxed{3 | 20} } } &&&&& l & {} \rlap{ \kern-1em \xrightarrow [ \textstyle \exists!\overline{\black G\lambda} \; \boxed{4 | 22} ] {\kern14em} } &&& \rightadj{ \big(\rightcat{(\black G Q)\lim_\calR}\big) R } \rlap{ \; \boxed{ \rightcat 3 \rightadj R | \rightcat{20} \rightadj R } } \\ \end{array} } } \]
The following combines (an external diagram) with (four internal diagrams) in (one large display).
The created limits are colored $\red{\text{red}}$.
\[ \boxed{ \begin{array} {c|ccccc|ccccccc|ccccc} 1& \forall \; \boxed{6 | 130} \; {\rightcat r} \leftcat{[\kappa]} & {} \rightcat{ \rlap{ \kern-2em \xrightarrow[\kern26em]{\textstyle \forall\rho \; \boxed{7 | 131}} } } &&& G = \rightcat{ (\black G Q) } \leftcat{ [{\black G} \lambda] } \; \boxed{1 | 12} & \kern0em & & && && && & \rightcat{\boxed{6Q | 30}} & \rightcat r & \rightcat{ {} \rlap{ \kern-1em \xrightarrow [\kern26em] { \textstyle \rho\rightcat Q \; \boxed{7Q | 31}} } } &&& \black G\rightcat Q \rlap{\; \rightcat{\boxed{2}}} & \kern2em \\ 2& & \leftcat{ \llap{ \exists! \; \boxed{8' | 132} \; \bar\rho} \searrow } & & \rightadj{ \nearrow \rlap{\pi_{\black G} \; \rightcat{\boxed{3' | 121}} } } &&& \rightcat{ {} \rlap{ \xrightarrow[\textstyle \kern8em \boxed{0 | 11} \kern8em]{\textstyle Q = r} } } & && && && & & & \rightcat{ \llap{ \boxed{8 | 32} \; \exists!\overline{\rho Q} } \searrow } & & \rightcat{ \nearrow \rlap{\pi_{\black G\rightcat Q} \; \boxed{3 | 21} } } & \\ 3& && \smash{ \raise7ex{ \red{ \boxed{ \textstyle \text{limit in } \leftcat{ (l \downarrow {\rightadj R}) } \over { \textstyle {\black G}\lim_{ \leftcat{ (l \downarrow {\rightadj R}) } } = \atop \textstyle \rightcat{ \big ((\black G Q)\lim_\calR\big) } \leftcat{ \Big[\overline{\black G \lambda}\Big] } } } } } } & \rightadj{ \boxed{5 | 120} } &&&& &&&& & & & &&& \smash{ \raise4ex{ \rightcat{ \boxed{ \textstyle \text{limit in } \calR \over \textstyle (\black G\rightcat Q)\lim_\calR = \leftcat l\leftadj L } } } } & \rightcat{\boxed{3 | 20}} & \\ \hline 4& && \leftcat{ (l \downarrow {\rightadj R}) } & &&& \calJ & \xrightarrow[\smash{\textstyle \kern1em \boxed{1 | 12} \kern1em}]{\textstyle G} & \leftcat{ (l \downarrow {\rightadj R}) } \rlap{ \smash{ \lower3ex{ \kern-2.5em \boxed{0 | 10} } } } & \rightcat{ \xrightarrow[\textstyle \kern1em \boxed{0 | 11} \kern1em]{\textstyle Q = r} } & \rightcat\calR & & & & &&& \rightcat\calR && \\ 5& && \Bigg\downarrow &&&& && \leftcat{ \llap ! \Bigg\downarrow } & \leftcat{ \llap{\boxed{0 | 11} \; \lambda} {\Bigg\Uparrow} } & \rightadj{ \Bigg\downarrow \rlap{ R \; \boxed{0} } } &&&&&&& \rightadj{ \Bigg\downarrow \rlap{ R \; \boxed{0} } } \\ 6& && \leftcat{ (l \downarrow \calL) } &&&& && 1 & \leftcat{ \xrightarrow[\textstyle \boxed{0 | 1} \; l]{} } & \leftcat\calL && & && && \leftcat\calL & \\ \hline 7& \forall \; \boxed{6' | 130'} \; \leftcat{ ({\rightcat r}{\rightadj R}) [\kappa] } & {} \rightcat{ \rlap{ \kern-1em \xrightarrow[\kern27em]{\textstyle \forall\rho \; \boxed{7 | 131} } } } &&& \leftcat{ (\black G\rightcat Q\rightadj R) [{\black G} \lambda] \rlap{\; \boxed{2 | 11}} } & &&& &&&&&& \leftcat{ \boxed{ \rightcat{6Q}\rightadj{R} | \rightcat{30}\rightadj{R} } } & \rightcat r\rightadj R & {} \rlap{ \kern-1em \xrightarrow [\kern25em] { \textstyle \rightcat{\rho Q}\rightadj R \; \boxed{ \rightcat{7Q}\rightadj R | \rightcat{31}\rightadj R } } } &&& \black G\rightcat Q\rightadj R \rlap{\; \leftcat{\boxed{2 \rightadj{R}}}} \\ 8& & \leftcat{ \llap{ \exists! \; \boxed{ {8'\rightadj R | 32 \rightadj R} } \; {\bar\rho}' } \searrow } && \rightadj{ \nearrow \rlap{ {\rightcat\pi}_{\black G\rightcat Q} \rightadj R \; \leftcat{ \boxed{ {\rightcat 3 \rightadj R | \rightcat{21} \rightadj R} } } } } &&& \rightcat{ {} \rlap{ \xrightarrow [\textstyle \kern8em \boxed{0 | 11} \kern8em] {\textstyle Q = r} } } & && && && & \leftcat{ \llap{\boxed{6\lambda} \; \kappa} \nearrow } & \leftcat{ {} \rlap{ \kern0em \black G\lambda \nearrow \kern0em \boxed{2 | \black G 11} } } &&& \rightcat{ \nearrow \rlap{ \rightcat{ \pi_{\black G Q} } \rightadj{R} \; \leftcat{ \boxed{ \rightcat{3}\rightadj{R} | \rightcat{21}\rightadj R } } } } & \\ 9& && \smash{ \raise2ex{ \red{ \boxed{ \textstyle \text{limit in } \leftcat{ (l \downarrow \calL) } \over { \textstyle \leftcat{ \Big( \rightcat{\big((\black G Q)\lim_\calR\big)} {\rightadj R} \Big) } \leftcat{\Big[\overline{\black G \lambda}\Big]} } } } } } & \rightadj{ \boxed{5' | 120'} } &&&& && && \kern1em & \kern1em & \leftcat{ \llap{\boxed{0 | 1}\;} l } & \leftcat{ {} \rlap{ \kern-1em \xrightarrow [ \textstyle \exists!\overline{\black G\lambda} \kern1em \boxed{4 | 32} ] {\kern9em} } } &&& \smash{ \raise0ex{ \leftcat{ \boxed{ \textstyle \text{limit in } \calL \over \textstyle \rightcat{ \big((\black G\rightcat Q)\lim_\calR\big) \rightadj R } = \leftcat l\leftadj L\rightadj R} } } } & \leftcat{ \boxed{ \rightcat 3\rightadj R | \rightcat{20}\rightadj R } } & \\ \end{array} } \]
Now suppose we have a functor $\boxed{ \boxed{1 | 12} \; G : \calJ \to \leftcat{(l \downarrow {\rightadj R})} }$ (see the center diagram in the large display above).
We form the components of $\boxed{1 | 12} \; G$ by composition, namely
a functor $\rightcat{ \boxed{ \boxed{2} \; {\black G}Q : {\black\calJ} \to \calR } }$ into $\rightcat\calR$, and
a natural transformation (a cone) $\leftcat{ \boxed{ \boxed{2} \; {\black G}\lambda : l \Rightarrow {\black G}{\rightcat Q}{\rightadj R} : {\black\calJ} \to \calL } }$ in $\leftcat\calL$.
Suppose now that (the functor $\rightcat{ \boxed{2} \; {\black G}Q : {\black\calJ} \to \calR}$) has
a limiting cone $\rightcat{ \boxed{ \textstyle \boxed{3 | 21} \; \pi_{\black G Q} : { \boxed{3|20} \; ({\black G}Q)\lim_\calR } \Rightarrow {\black G} Q : {\black\calJ} \to \calR } }$ in $\rightcat\calR$ (at the upper right of the above large display),
and that (that limiting cone in $\rightcat\calR$) is preserved by $\rightadj R$ (so we are now in the lower right of the above display).
Then there exists a unique (mediating arrow $\leftcat{ \boxed{\textstyle \boxed{4 | 22} \; {\overline{{\black G}\lambda}} : l \to \rightadj{ \big(\rightcat{(\black G Q)\lim_\calR}\big) R } } }$ in $\leftcat\calL$),
which is also (a <u>morphism of cones in $\leftcat\calL$</u>)
$\leftcat{ \boxed{ \boxed{4 | 22} \;{\overline{{\black G}\lambda}} : {\black G}\lambda \to { \rightcat{ \pi_{\black G Q} } }{\rightadj R} } }$ (in the lower right of the display).
But, and here is a crucial but somewhat tricky, perhaps subtle, point,
(that same data) may.be viewed as (<u><i>a single cone</i> in $(l\downarrow \rightadj R)$</u>)
$\rightadj{ \boxed{\textstyle \boxed{5 | 121} \; \pi_{\black G} : \boxed{5|120} \; \rightcat{\big((\black G Q)\lim_\calR\big)} \leftcat{[\overline{\black G \lambda}]} \Rightarrow \leftcat{ G : \calJ \to (l\downarrow \rightadj R) } } }$ (shown at the upper left of the above display).
Thus the same arrow $\leftcat{ \boxed{\textstyle \boxed{4 | 22} \; { \overline{{\black G}\lambda} } } }$ plays three different roles:
mediating arrow $\leftcat{ \boxed{\textstyle \boxed{4 | 22} \; {\overline{{\black G}\lambda}} : l \to \rightadj{ \big(\rightcat{(\black G Q)\lim_\calR}\big) R } } }$ in $\leftcat\calL$
morphism of cones $\leftcat{ \boxed{ \textstyle \boxed{4 | 22} \; { \overline{ {\black G}\lambda } } : \leftcat l [{\black G}\lambda] \to \Big( \rightadj{ \big( \rightcat{(\black G Q)\lim_\calR} \big) R } \Big) [{ \rightcat{ \pi_{\black G Q} } }{\rightadj R} ] } }$ in $(\Delta \downarrow \ulcorner G\rightcat Q\rightadj R\urcorner)$
a component of the object $ \rightadj{ \boxed{ \textstyle \boxed{120} \; \rightadj{ \big( \rightcat{(\black G Q)\lim_\calR} \big) } \leftcat{ \big[ \, \overline{\black G \lambda} \, \big] } } } $ in $\leftcat{ (l \downarrow \rightadj R) }$
<hr />
Same data but different notation:
\[ \leftcat{ \begin{array} {} && 1 \\ & \llap{!} \nearrow & \red{ \Bigg\downarrow \rlap{ \kern-2.8em \boxed{(G_i[\lambda_i])\lim} } } & \rightcat{ \searrow \rlap{ G_i\lim } } \\ \calJ & \xrightarrow[\kern3em]{\textstyle {\rightcat G}_i[\lambda_i]} & \boxed{(l \downarrow {\rightadj R})} & \rightcat{ \xrightarrow[\kern3em]{\textstyle Q = r} } & \rightcat\calR \\ && \Bigg\downarrow & \buildrel \textstyle \lambda \over \Longrightarrow & \rightadj{\Bigg\downarrow \rlap R} \\ && 1 & \xrightarrow[\textstyle l]{} & \calL \\ \end{array} } \]
\[ \leftcat{ \boxed{ \begin{array} {cccccc c|cccccc c|cccccc} {} \rlap{ \kern7em (l\downarrow \rightadj R) \kern.5em \rightcat{ \xrightarrow[\kern20em]{\textstyle Q = r} } } &&&&& && {} \rlap{ \kern6em \rightcat\calR \kern.5em \rightadj{ \xrightarrow[\kern27em]{\textstyle R} } } &&&&&&& {} \rlap{ \kern11em \calL } \\ \hline & \forall \rightcat r[\kappa] & \rightcat{ {} \rlap{ \kern-1em \xrightarrow[\kern12em]{\textstyle \forall \rho_i} } } &&& {\rightcat G}_i[\lambda_i] & \kern1em & \kern1em & \rightcat{\forall r} & \rightcat{ {} \rlap{ \kern-1em \xrightarrow[\kern10em]{\textstyle \forall \rho_i} } } &&& \rightcat{ G_i} & \kern1em & \kern1em &&& {\rightcat r}{\rightadj R} & \rightcat{ {} \rlap{ \kern-1em \xrightarrow [\kern11em] {\textstyle \rho_i \rightadj R } } } &&& \rightcat{G_i} {\rightadj R} & \\ && \rightcat{ \llap{\exists!\rho} \searrow } && \rightcat{ \nearrow \rlap{\pi_i} } & &&& & \rightcat{ \llap{\exists!\rho} \searrow } && \rightcat{ \nearrow \rlap{\pi_i} } &&&&& \llap{\kappa} \nearrow && \llap{\lambda_i} \nearrow && \rightcat{ \nearrow \rlap{\pi_i \rightadj R} } \\ &&& \rightcat{ G_i\lim}[\lambda] && && &&& \rightcat{G_i\lim} &&&&& l & {} \rlap{ \kern-1em \xrightarrow [\textstyle \exists!\lambda] {\kern8em} } &&& \rightcat{G_i\lim}\rightadj R \\ \end{array} } } \]
We start with a functor ${\rightcat G_i[\leftcat\lambda_i]} : \calJ \to \leftcat{(l\downarrow {\rightadj R})}$, so $\rightcat{ G : \calJ \to \calR}$ is a functor landing in $\rightcat\calR$, while $\{ \leftcat{ \lambda_i : l \rightarrow {\rightcat G_i \rightadj R} } \}_{i\in\calJ}$ is a cone in $\leftcat\calL$.
Suppose $\rightcat G$ has a.limit $\rightcat{\pi_i : G\lim \to G_i}$ in $\rightcat\calR$ which is preserved by $\rightadj R$.
In these circumstances the limit $\rightcat{\pi_i : G\lim \to G_i}$ in $\rightcat\calR$ lifts to a limit in $\leftcat{(l\downarrow \rightadj R)}$.
\[ \leftcat{ \boxed{ \begin{array} {ccccccc c|cccccc c|cccccc} {} \rlap{ \kern8em \calL \kern8em \rightadj{ \xleftarrow[\kern4em]{\textstyle R} } } &&&&&&&&& {} \rlap{ \kern5em \rightcat{ \calR \kern5em \xleftarrow[\kern4em]{\textstyle Q = r} } } &&&&&&& {} \rlap{ \kern4.5em (l\downarrow \rightadj R) } &&&&& test \\ \hline && {\rightcat r}{\rightadj R} & \rightcat{ {} \rlap{ \kern-1em \xrightarrow [\kern11em] {\textstyle \rho_i \rightadj R } } } &&& \rightcat{G_i} {\rightadj R} & \kern1em & \kern1em & \rightcat{\forall r} & \rightcat{ {} \rlap{ \kern-1em \xrightarrow[\kern10em]{\textstyle \forall \rho_i} } } &&& \rightcat{ G_i} & \kern1em & \kern1em & \forall \rightcat r[\mu] & \rightcat{ {} \rlap{ \kern-1em \xrightarrow[\kern12em]{\textstyle \forall \rho_i} } } &&& {\rightcat G}_i[\lambda_i] & test \\ & \llap{\kappa} \nearrow && \llap{\lambda_i} \nearrow && \rightcat{ \nearrow \rlap{\pi_i \rightadj R} } & &&& & \rightcat{ \llap{\exists!\rho} \searrow } && \rightcat{ \nearrow \rlap{\pi_i} } &&& && \rightcat{ \llap{\exists!\rho} \searrow } && \rightcat{ \nearrow \rlap{\pi_i} } && test \\ l & {} \rlap{ \kern-1em \xrightarrow [\textstyle \exists!\lambda] {\kern8em} } &&& \rightcat{G_i\lim}\rightadj R && &&& && \rightcat{G_i\lim} &&& && && \rightcat{ G_i\lim}[\lambda] &&& test \\ \end{array} } } \]
Below this line is just excess material, basically earlier draft stuff that I am keeping around in case I want to use them again.
\[ \leftcat{ \boxed{ \begin{array} {cc|ccc|cccc} \calJ \rlap{ \xrightarrow[\kern4.5em \boxed{12} \kern4.5em] {\textstyle \boxed{ \black G = \black G \rightcat Q [\black G \lambda]} } } & \kern0em & \kern0em & (l\downarrow\calL) \rlap{ \lower3.2ex{ \kern-2.3em \boxed{10} } } \rlap{ \rightcat{ \xrightarrow[ \textstyle \kern3.6em \boxed{11} \; \text{faithful} \kern3.6em ]{\textstyle Q=r} } } & & & & &\calL & \\ \\ \hline j &&& j\black G = j \big( \black G \rightcat Q [\black G \lambda] \big) = j \black G \rightcat Q [j \black G \lambda] &&&&& j\black G\rightcat Q \\ &&& &&&& \llap{j \black G \lambda} \nearrow & \\ \llap\iota \Bigg\downarrow &&& \llap{\iota \black G = {} } \Bigg\downarrow \rlap{ {} = \iota \black G \rightcat Q } & && l & \iota \black G & \Bigg\downarrow \rlap{\iota \black G \rightcat Q} & \\ &&&& && & \llap{j' \black G \lambda} \searrow & \\ j' &&& j' \black G = j' \big( \black G \rightcat Q [\black G \lambda] \big) = j' \black G \rightcat Q [j' \black G \lambda] &&&&& j' \black G\rightcat Q \\ \end{array} } } \]
