<h3> What is a greatest element?</h3>
Here we gather together some basic results about the ubiquitous and highly significant "terminal objects" in categories.
Perhaps most important are several alternative characterizations of (the "terminal object" ${\rightadj\top}_\calC$) in (an arbitrary category $\calC$) see.
The key result can be described as this:
\[ \boxed{ \displaystyle {\rightcat 0}_\calC \rightadj\limit \cong \top_\calC \cong {\leftcat 1}_\calC \leftadj\colimit } \; .\]
This references two functors into $\calC$, the least, ${\rightcat 0}_\calC$, and the greatest, ${\leftcat 1}_\calC$
( this situation should be compared to (the power set $XP$) of (an arbitrary set $X$), which has two extremal elements, (the empty, least, subset $\emptyset$), and (the greatest subset, $X$ itself) ):
\[ \begin{array} {} &&&&& \kern16em & \rightcat{ \hom {{{\leftadj !}_{\rightcat 0}}\leftcat c} {[0,\black\calC]} {0_\calC} } & \cong & 1 \\ \rightcat 0 & \rightcat{ \xrightarrow[\kern8em]{\textstyle 0_\calC \text{ empty functor}} } & \calC && && \llap{ \text{defn. } \rightadj( {\rightcat 0}_\calC \rightadj\limit \rightadj) \; } {\wr\Vert} && {\wr\Vert} \rlap{ \text{ defn. } {\top_\calC} } \\ \llap{ {\leftadj !}_{\rightcat 0} } \leftadj{\Bigg\downarrow} & \rightadj{ \llap\epsilon \Bigg\Uparrow \rlap{ { \textstyle \text{empty} \atop \textstyle \text{cone} } } } & \Bigg\Vert &&&& \hom {\leftcat c} {\leftcat\calC} { \rightadj( {\rightcat 0}_\calC \rightadj\limit \rightadj) } && \hom {\leftcat c} {\leftcat\calC} {\top_\calC} \\ 1 & \xrightarrow[\smash{\raise2.4ex{\displaystyle \top_\calC}}]{\kern8em} & \calC \\ \rightadj{ \llap{ !_{\leftcat\calC} } \Bigg\uparrow } & \leftadj{ \llap\eta \Bigg\Uparrow \rlap ! } & \Bigg\Vert & &&& \rightcat{ \hom { \leftadj ( \leftcat{1_\calC} \leftadj\colimit \leftadj ) } \calC c } \\ \leftcat\calC & \leftcat{ \xrightarrow[\textstyle 1_\calC \text{ identity functor}]{} } & \calC && && \llap{ \text{defn. } \leftadj( \leftcat{1_\calC} \leftadj\colimit \leftadj) \; } {\wr\Vert} \\ &&&&& & \leftcat{ \hom {1_\calC} {[\calC,\black\calC]} {{{\rightadj !}_\calC}\rightcat c} } \\ \end{array} \]
The isomorphism involving $\rightadj\limit$ is proved by the bijections at top right above.
The isomorphism involving $\leftadj\colimit$ is proved in the diagrams below;
the two on the left show that
(a terminal, denoted $\rightadj{\top[\leftadj !]}$) is (a colimit of the identity endofunctor $\leftcat{1_\calC}$),
while the two on the right show that
(a colimit, denoted $\leftadj{c_\infty[\iota]}$, of the identity endofunctor $\leftcat{1_\calC}$) is (a terminal in $\calC$).
\[ \boxed{ \begin{array} {ccccccccc|cccc} & {} \rlap{ \kern0em \rightadj{ \top[\leftadj !] } \text{ is initial in } ({\leftcat{1_\calC}} \downarrow \Delta), \text{ thus is } \leftadj{ (\leftcat{1_\calC} \colimit) [\iota] } } &&&& &&& && {} \rlap{ \kern2em \leftadj{ ( c_\infty = \leftcat{1_\calC} \colimit) [\iota] } \text{ is } \rightadj{ \top [\leftadj !] } } &&& \\ \\ \hline \kern5em & \rightadj\top & \leftcat{ \xrightarrow[\kern3em]{ \textstyle \lambda_{\rightadj\top} } } & \rightcat d & \kern8em & \rightadj\top & \leftcat{ \xrightarrow[\kern3em]{ \textstyle \rightcat f } } & \rightcat d & \kern2em & \kern4em & \leftadj{c_\infty} & \leftadj{ \xrightarrow[\kern3em]{ \textstyle \iota_{c_\infty} } } & \leftadj{c_\infty} & \kern2em & \kern2em & \leftadj{c_\infty} & \leftadj{ \xrightarrow[\kern3em]{ \textstyle \iota_{c_\infty} = 1_{c_\infty} } } & \leftadj{c_\infty} \\ & \leftadj{ \llap{ !_{\leftcat c} } \Big\uparrow } \rlap{ \kern.5em \leftcat\lambda_{ \leftadj!_{\leftcat c} } } & \leftcat{ \nearrow \rlap{\lambda_c} } &&& \leftadj{ \llap{ \rightadj{ {\leftadj !}_\top = 1_\top } } \Big\uparrow } \rlap{ \kern.5em \rightcat f } & \leftcat{ \nearrow \rlap{{\leftcat\lambda}_{\rightadj\top}} } & & && \leftadj{ \llap{ \iota_{\leftcat c} } \Big\uparrow \rlap{ \kern.5em \iota_{\iota_{\leftcat c}} } } & \leftadj{ \nearrow \rlap{ \iota_{\leftcat c} } } &&&& \leftcat{ \llap{ \forall f } \Big\uparrow \rlap{ \kern.5em {\leftadj\iota}_f } } & \leftadj{ \nearrow \rlap{ \iota_{\leftcat c} } } \\ & \leftcat c &&& & \rightadj\top &&& && \leftcat{ \llap\forall c } &&& && \leftcat c \\ \hline {} \rlap{ \kern-3em { { \textstyle \text{if } {\leftcat\lambda} : \leftcat{1_\calC \Rightarrow {\rightcat d}{\black\Delta} \text{ ($\lambda$ is a cocone from $\leftcat{1_\calC}$ to $\rightcat d$)} \, , } \atop \textstyle \text{then, in particular, $\leftcat{\forall c}$ the above ${\leftcat\lambda}_{ {\leftadj !}_{\leftcat c} }$ commutes,} } \atop \textstyle \text{i.e. $\leftcat\lambda$ factors through $\leftadj !$ by ${\leftcat\lambda}_{\rightadj\top}$} } } &&&& {} \rlap{ \kern4em { { \textstyle \text{if } \rightcat{ f : {\rightadj\top}[{\leftadj !}] \to {\rightcat d}[{\leftcat\lambda}] } \atop \textstyle \text{the above commutes;} } \atop \textstyle \text{hence $\rightcat f = {\leftcat\lambda}_{\rightadj\top}$ } } } &&&&& {} \rlap{ \kern-2em { { \textstyle \text{because $\leftadj\iota$ is a cocone, $\leftadj{ \iota : \leftcat{1_\calC} \Rightarrow c_\infty {\black\Delta} }$, } \atop \textstyle \text{$\leftcat{\forall c}$ the above $\leftadj{\iota_{ \iota_{\leftcat c}}}$ commutes;} } \atop \textstyle \text{hence, by the colimit property, $\leftadj{ \iota_{c_\infty} = 1_{c_\infty} }$} } } &&&&& {} \rlap{ \kern0em { { \textstyle \text{because $\leftadj\iota$ is a cocone,} \atop \textstyle \text{$\leftcat{\forall f}$ the above ${\leftadj\iota}_f$ commutes;} } \atop \textstyle \text{hence $\leftcat{\forall f}$ $\leftcat{ f = {\leftadj\iota}_c }$} } } \\ \end{array} } \]
In words:
To show $\rightadj{\top[{\leftadj !}]}$ is initial in the comma category $({\leftcat{1_\calC}} \downarrow \Delta)$,
let $\leftcat{ (\lambda : {1_\calC} \Rightarrow {\rightcat d} ) }$ be an arbitrary cocone over $\leftcat{1_\calC}$.
Since $\leftcat\lambda$ is a cocone, the triangle at left commutes.
Thus $\leftcat\lambda$ factors through $\leftadj !$ by ${\leftcat\lambda}_{\rightadj\top}$.
Further, recall $\rightadj{ {\leftadj !}_\top = 1_\top }$ by uniqueness in $\hom {\rightadj\top} \calC {\rightadj\top}$.
Thus if $\rightcat{ f : {\rightadj\top}[{\leftadj !}] \to {\rightcat d}[{\leftcat\lambda}] }$, taking $\leftcat{ c = {\rightadj\top} }$ yields $\rightcat{ f = {{\leftcat\lambda}_{\rightadj\top}} }$.
That proves the left assertion in the box above.
<hr />
${\rightadj\top}[{\leftadj !}]$ is an <em>absolute colimit</em>.
(This is (the dual of) CWM Exercise III.4.3.)
This result may be expressed as saying that:
For any (functor $\leftcat{ F: \calC \to {\rightcat\calD} }$)
(the reflection adjunction $\leftadj{!_{\leftcat\calC}} \dashv \rightadj{\top_{\leftcat\calC}}$) induces, via pasting composition,
a "mates" bijection between 2-cells
$\leftcat{ ( \alpha : F \Rightarrow \leftadj{!_{\leftcat\calC}} {\rightcat d} ) }$ and $\rightcat{ ( \delta : \rightadj{\top_{\leftcat\calC}} {\leftcat F} \Rightarrow {\rightcat d} ) }$,
that is, of types indicated in the 2-diagram below:
\[ \begin{array} {} && \rightcat\calI & \rightcat{ {} \rlap{ \kern-1em \xrightarrow[\kern8em]{} } } &&& \rightcat\calI & \rightcat{ {} \rlap{ \kern-1em \xrightarrow[\kern8em]{\textstyle d} } } &&& \rightcat\calD \\ & \leftadj{ \llap !_{\leftcat\calC} \nearrow } & \smash{ \leftadj{ \llap ! \Big\Uparrow \rlap\eta } } & \rightadj{ \searrow \rlap{ \kern-.5em \top_{\leftcat\calC} } } & \smash{ \rightadj{ \Big\Vert \rlap\epsilon } } & \leftadj{ \nearrow \rlap{ \kern-1em !_{\leftcat\calC}} } & \smash{ \leftadj{ \llap ! \Big\Uparrow \rlap{ \eta \kern0em \leftcat{ \Bigg\Uparrow\rlap{\kern-.7em\alpha} } } } } & \rightadj{ \searrow \rlap{ \kern-.5em \top_{\leftcat\calC} } } & \smash{ \rightcat{ \Big\Uparrow \rlap\delta } } & \leftcat{ \nearrow \rlap F } \\ \leftcat\calC & \leftcat{ {} \rlap{ \kern-1em \smash{ \xrightarrow[\kern8em]{\smash{}} } } } &&& \leftcat\calC & \leftcat{ {} \rlap{ \kern-1em \smash{ \xrightarrow[\kern8em]{\smash{}} } } } &&& \leftcat\calC \\\end{array} \]
The full-up mates bijection involves a morphism between two adjunctions, (one on the left) and (one on the right).
In the case above,
(the left adjunction is $\leftadj{!_{\leftcat\calC}} \dashv \rightadj{\top_{\leftcat\calC}}$),
while (the one on the right is the trivial adjunction $1_\calD \dashv 1_\calD$).
From that POV, the mates bijection would be diagrammed as:
\[ \begin{array} {} && \rightcat\calI & \rightcat{ {} \rlap{ \kern-1em \xrightarrow[\kern8em]{} } } &&& \rightcat\calI & \rightcat{ {} \rlap{ \kern-1em \xrightarrow[\kern8em]{\textstyle d} } } &&& \rightcat\calD \\ & \leftadj{ \llap !_{\leftcat\calC} \nearrow } & \smash{ \leftadj{ \llap ! \Big\Uparrow \rlap\eta } } & \rightadj{ \searrow \rlap{ \kern-.5em \top_{\leftcat\calC} } } & \smash{ \rightadj{ \Big\Vert \rlap\epsilon } } & \leftadj{ \nearrow \rlap{ \kern-1em !_{\leftcat\calC}} } & \smash{ \leftadj{ \llap ! \Big\Uparrow \rlap{ \eta \kern0em \leftcat{ \Bigg\Uparrow\rlap{\kern-.7em\alpha} } } } } & \rightadj{ \searrow \rlap{ \kern-.5em \top_{\leftcat\calC} } } && \smash{ \rightcat{ \Bigg\Uparrow \rlap\delta } } & & \rightadj{ \searrow \rlap{1_\calD} } \\ \leftcat\calC & \leftcat{ {} \rlap{ \kern-1em \smash{ \xrightarrow[\kern8em]{\smash{}} } } } &&& \leftcat\calC & \leftcat{ {} \rlap{ \kern-1em \smash{ \xrightarrow[\kern8em]{\smash{}} } } } &&& \leftcat\calC & \leftcat{ {} \rlap{ \kern-1em \xrightarrow[\textstyle F]{\kern8em} } } &&& \leftcat\calD \\ \end{array} \]
<hr />
This situation may be internally diagrammed in $\leftcat\calC$ and $\rightcat\calD$ as follows.
Given a cocone $\leftcat{ ( \alpha : F \Rightarrow \leftadj{ !_{\leftcat\calC} } {\rightcat d} ) }$ in $\rightcat\calD$,
then, for each $\leftcat c$ in $\leftcat\calC$, we have the following:
\[ \boxed{ \begin{array} {ccc|cccc} & \leftcat{ \calC \rlap{ {} \xrightarrow[\kern6em]{\textstyle F} } } & \kern0em & \kern0em & \rightcat{ {} \rlap{ \kern1em \calD } } \\ \hline & {\rightadj\top}_{\kern-.2em \leftcat\calC} &&& {\rightadj\top}_{\kern-.2em \leftcat\calC} \, {\leftcat F} & \rightcat{ \xrightarrow{\textstyle {\rightadj\top}_{\kern-.2em \leftcat\calC} \, \leftcat\alpha} } & \rightcat d \\ & \leftadj{ \llap{ \exists ! \, !_{\leftcat c} } {\Big\uparrow} } \rlap{\leftcat\gamma} &&& \leftadj{ \llap{!_{\leftcat c} \, {\leftcat F}} {\Big\uparrow} } \rlap{ \kern.5em {\leftadj !}_{\leftcat c} \, {\leftcat\alpha} } & \leftcat{ \nearrow \rlap{ {\leftcat c} \, \alpha } } \\ & \leftcat{ \llap{\forall} c } &&& {\leftcat c} \, {\leftcat F} \\ \end{array} } \]
This is in fact a special case of the more general Yoneda situation.
To clarify the relation, I will keep the same notation, but temporarily
DROP THE ASSUMPTION THAT $\rightadj\top$ IS A TERMINAL OBJECT.
So consider the situation
\[ \boxed{ \begin{array} {ccc|cccc} & \leftcat{ \calC \rlap{ {} \xrightarrow[\kern7.5em]{\textstyle F} } } & \kern0em & \kern0em & \rightcat{ {} \rlap{ \kern2em \calD } } \\ \hline & {\rightadj\top}_{\kern-.2em \leftcat\calC} &&& {\rightadj\top}_{\kern-.2em \leftcat\calC} \, {\leftcat F} & \rightcat{ \xrightarrow{\textstyle \delta} } & \rightcat d & \kern1em \\ & \leftcat{ \llap{\forall\gamma} {\Big\uparrow} } &&& \leftcat{ \llap{\gamma F} {\Big\uparrow} } & \leftcat{ \nearrow \rlap{ {\leftcat\gamma} \, \rightcat{(\beta = \hat\delta)} } } \\ & \leftcat{ \llap{\forall} c } &&& {\leftcat c} \, {\leftcat F} \\ \end{array} } \]
This gives the direction $\rightcat\delta \mapsto \rightcat{(\beta = \hat\delta)}$ of the classic Yoneda bijection:
\[ \begin{array} {} \rightcat\delta & \mapsto & \rightcat{(\beta = \hat\delta)} \\ \rightcat{ \hom {\rightadj\top \leftcat F} \calD d } & \buildrel \text{Yon} \over \cong & \hom { \leftcat{ \hom - \calC {\rightadj\top} } } {[\leftcat{\calC\op},\Set]} { \rightcat{ \hom { \leftcat{- F} } \calD d } } \\ && \leftcat\gamma \mapsto \rightcat{ \big( {\leftcat\gamma}\beta = \leftcat{\gamma F} \cdot \delta \big) } \\ && {} \rlap{ \kern-10em \text{ $\rightcat{(\beta = \hat\delta)}$ may be viewed as the $\leftcat{\calC\op}$-orbit in $\rightcat\calD$ of $\rightcat\delta$. } } \\ \end{array} \]
The earlier result comes from reinstating the assumption that $\rightadj\top$ is terminal,
and noting that in that case, for a given $\leftcat{c \in \calC}$, there is only one choice for $\leftcat{ \gamma : c \to \rightadj\top }$, namely ${\leftadj !}_{\leftcat c}$, hence here $\rightcat{(\beta = \hat\delta)}$ is a function of $\leftcat{c \in \calC}$ alone, which we denote $\alpha$.
This will be further illustrated below.
<hr />
Although not necessary to understand the above situation,
it is of interest to me at least to compare the above situation
to that of the Yoneda lemma.
There is a certain external diagram that is often used in proving the Yoneda lemma, which in this case instantiates to:
\[ \begin{array} {} \rightadj{1_\top} & \leftcat\in & \hom {\rightadj\top} {\leftcat\calC} {\rightadj\top} & {} \rlap{ \kern-1em \xrightarrow[\kern2em]{ {\rightadj\top} {\beta} } } & \rightcat{ \hom {{\rightadj\top} {\black F}} \calD d } & \rightcat\ni & \rightcat{ \rightadj{1_\top} \, ( {\rightadj\top} {\beta} ) = \delta } \\ && \leftcat{ \llap{ \hom \gamma \catC {\rightadj\top} } \Big\downarrow } & {\leftcat\gamma} \beta & \rightcat{ \Big\downarrow \rlap{ \hom {{\leftcat\gamma} {\black F}} \calD d } } \\ \llap{ {\leftadj !}_{\leftcat c} = {} } \leftcat\gamma & \leftcat\in & \hom {\leftcat c} {\leftcat\calC} {\rightadj\top} & {} \rlap{ \kern-1em \xrightarrow[ {\leftcat c} {\beta} ]{\kern2em} } & \rightcat{ \hom {{\leftcat c} {\black F}} \calD d } & \rightcat\ni & \rightcat{ {\leftcat\gamma} (\leftcat c \beta) = \leftcat{\gamma F} \cdot \delta } \\ \end{array} \]
Here we consider how several bijections may be composed,
demonstrating two alternative ways, mates versus Yoneda, of proving that $\leftadj{ !_{\leftcat\calC} }$ is an absolute colimit:
\[ \begin{array} {} \leftcat\alpha && \rightcat{ \delta = \check\beta } && \rightcat{ \beta = \hat\delta } \\ \leftcat{ \hom F { [\leftcat\calC,\rightcat\calD] } { \leftadj{ !_{\leftcat\calC} } {\rightcat d} } } & \buildrel \text{mates} \over \cong & \rightcat{ \hom {\rightadj\top \leftcat F} \calD d } & \buildrel \text{Yon} \over \cong & \hom { \leftcat{ \hom - \calC {\rightadj\top} } } {[\leftcat{\calC\op},\Set]} { \rightcat{ \hom { \leftcat{- F} } \calD d } } \\ {} \rlap{ \kern-4em \text{cocone from $\leftcat F$ to $\rightcat d$} } &&&& {\wr\Vert} \rlap{ \text{ if } \rightadj\top \text{ is terminal} } \\ &&&& \hom { \leftadj{ !_{\leftcat{\calC\op}} } 1 } {[\leftcat{\calC\op},\Set]} { \rightcat{ \hom { \leftcat{- F} } \calD d } } \\ &&&& {\wr\Vert} \rlap{ \text{ remark in proof of CWM Thm. V.4.1} } \\ &&&& \hom { \leftcat F } {[\leftcat\calC,\rightcat\calD]} { \leftadj{ !_{\leftcat\calC} } \rightcat d } \\ &&&& \leftcat\alpha \end{array} \]
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