Limit, colimit, initial, terminal, adjoint

We show various relations between the concepts mentioned in the title.

We begin by recalling one of the definitions of colimit.

If $F : \calJ \to \calC$ is a functor, then its colimit, if it exists, 

is a left extension diagram in the 2-category $\Cat$:

$$\begin{array}{} && \llap{ (\text{ unit category} = {}) \kern.5em } {\mathcal I}   \\    & \llap ! \nearrow  & \leftadj{ \Big \Uparrow \rlap \iota }   & \leftadj\searrow \rlap{\leftadj{\text{colimit }} F}    \\ \mathcal J  & {} \rlap{ \kern-1em \xrightarrow[\textstyle F]{\kern7em} }  &&& \mathcal C  \\   \end{array}$$

Now specialize to ($\mathcal J = \mathbf 0$, the empty category), 

and ($\boxed{ F = \bigcirc : \mathbf  0 \to \calC }$ the unique (empty) functor).

Then (the above left extension diagram for $(\text{colimit }F)$) specializes to

$$\begin{array}{} &&  \llap{ (\text{ unit category} = {}) \kern.5em } {(\mathcal I = \mathbf 1)}  \\    & \llap ! \nearrow  & \leftadj{ \Big \Uparrow \rlap \iota }   & \leftadj\searrow \rlap{ \leftadj{\text{colimit }} \bigcirc }   \\     \llap{ (\text{empty category } = {} ) \kern.5em  } {\mathbf 0} &  {} \rlap{    \kern-1em \xrightarrow[\textstyle \bigcirc]{\kern11em} }  &&& \mathcal C  \\  \end{array}$$

There is only one possible natural transformation out of (the empty functor $\bigcirc$), thus we have the bijection (*) in:

$$\boxed{     \begin{array}{}  \kern7.5em  & \hom \bigcirc  {[\mathbf 0, \calC]} {!c}  &  \buildrel \text{(*)} \over \cong  &  \mathbf 1  & \kern12em   \\  &  \llap{\text{definition (colimit $\bigcirc$)}} {\wr\Vert}    &&     {\wr\Vert} \rlap{\text{ definition (initial object = $\bot$)}}   \\  &   {} \rlap{ \kern-3.8em \hom {\text{(colimit $\bigcirc$)}} \calC c }   &&   \hom \bot \calC c   \\ \end{array}     }$$

Since this is true (for all $c \in \calC$), we have $\boxed{  \big( \text{colimit } (\bigcirc : \mathbf  0 \to \calC) \big) \cong \bot  }$, i.e., 

(an initial object $\bot$) is (a colimit of (the empty functor)). 

Note that the above proof only needed the bijection (*) and the definitions of colimit and initial object.

This generalizes the order-theoretic result that, in a preorder, 

 (a least upper bound, i.e. supreum, for the empty subset) is (a bottom).

For an example of non-uniqueness, 

consider a set with two or more elements with the indiscrete (chaotic) preorder, which is certainly not antisymmetric, thus is a preorder but not a partial order.

For such a preorder, every element is both a lub($\emptyset$) and a bottom.

As to the existence of <i>minimal</i> elements, <a href="https://en.wikipedia.org/wiki/Greatest_element_and_least_element">Wikipedia</a> gives two definitions, one for preorders and one for partial orders.

Per the preorder definition, which is the appropriate definition here, 

<i>EVERY</i> element is minimal.

Per the (more familiar) definition for partial orders (which makes sense even for preorders, even if it is not the proper definition in those cases), 

<i>NO</i> element is minimal.

<hr />

In addition to (initial objects) being (colimits of the smallest possible functor into $\mathcal C$, $\bigcirc : \mathbf 0 \to \mathcal C$),

(initial objects) are also (limits of the largest possible functor into $\mathcal C$,  the identity functor $1_{\mathcal C} : \mathcal C \to \mathcal C$), 

generalizing the fact that in pre-orders, bottoms are infima, i.e. greatest lower bounds, for the entire pre-order.

(A bottom $\leftadj\bot$ in $\mathcal C$) has (a unique arrow $\boxed{ \bigcirc_c : \leftadj\bot \to c }$) into (each object $c \in \calC$).

(Note that we use the same symbol, $\bigcirc$, both

externally, in $\CAT$, to denote (the unique arrow (a functor)) from (the initial object $\mathbf 0$) to (an arbitrary object, a category, $\mathcal C$) in $\CAT$, and 

internally, in $\calC$, to denote (the unique arrow) from (the initial object $\bot$) to (an arbitrary object $c$) in (a given category $\mathcal C$).)

These provide (the projection arrows) necessary to make $\leftadj\bot$ (a limit for the functor $1_{\calC}$).

Two key facts about $\bigcirc$ follow from (the uniqueness condition) in (the definition of initiality): since ($\leftadj\bot$ is initial), 

$\bullet$ There is one and only one endoarrow (the unique self-map) $\leftadj\bot \to \leftadj\bot$, 

thus $\boxed{  ( \bigcirc_{\leftadj\bot} = 1_{\leftadj\bot} ) : \leftadj\bot \to \leftadj\bot  }$.

$\bullet$ The family of arrows $\{\bigcirc_c : \leftadj\bot \to c\}_{c \in \calC}$ collectively form a cone $\boxed{ \bigcirc : {\leftadj\bot}\Delta \Rightarrow 1_{\calC} }$;

the transformation $\bigcirc$  is natural since $\bigcirc_c \gamma = \bigcirc_{c'}$ for any $\gamma : c \to {c'}$ in $\calC$.

$$\begin{array}{} && \leftadj\bot   \\   & \llap{\bigcirc_c} \swarrow  &&   \searrow \rlap{\bigcirc_{c'}}   \\    c & {}\rlap{ \kern-.5em \xrightarrow[\textstyle \gamma]{\kern7em} } &&& {c'}   \end{array}$$

Thus the family of arrows $\{\bigcirc_c : \leftadj\bot \to c\}_{c \in \calC}$ 

is closed under post-composition, i.e., is a one-sided ideal.

<hr />

To show ($\boxed{ {\leftadj\bot} [\bigcirc] = <\leftadj\bot, {\leftadj\bot}\Delta \buildrel \bigcirc \over \Rightarrow 1_{\calC}>}$ is a limit for $1_{\calC}$), 

we must show (${\leftadj\bot} [\bigcirc]$ is terminal in $(\Delta \downarrow \ulcorner 1_\calC \urcorner)$, the category of cones to $1_\calC$),

i.e. the comma category arising from the displayed cospan in $\CAT$:

$$\begin{array}{cc|cccccccc|c|cccc}  &&&&  &&  \boxed{ (\Delta \downarrow \ulcorner 1_\calC \urcorner) } \rlap{ \text{ ( = cones to $1_\calC$) } }     \\   &&&&    & \llap b \swarrow & & \searrow \llap !     \\   &&&&    \calC &&  \buildrel \textstyle \tau \over \Rightarrow && \mathcal I     \\   &&&&  & \llap{\Delta} \searrow  &   \CAT   &   \swarrow \rlap{ \ulcorner 1_\calC \urcorner }   \\   &&&&   &&  [\calC, \calC]  \\  \\    \hline    &&&& &&&&&&  \kern6em & {\leftcat b}   \\   {\leftcat b}[\tau] &&&&  & {\leftcat b} \Delta  &  \xrightarrow[\kern2em]{\textstyle \tau}  &  1_\calC  & &  &  &  &  \searrow \rlap{\tau_c}  & \rlap{\tau_\gamma} & \searrow \rlap{\tau_{c'}}    \\ \llap{ \text{(the generic arrow in $(\Delta \downarrow 1_\calC)$)} \kern2em \leftcat\beta} \Bigg\downarrow &&&&  &  \llap{\leftcat\beta  \Delta} \Bigg \downarrow  &  [\calC, \calC]  &  \Bigg\Vert  & &  &  &  \llap{\leftcat \beta} \Bigg \downarrow  &  \calC  &  c   &  \xrightarrow[\kern3em]{\gamma}  & c'   \\ {\leftcat{b'}}[\tau'] &&&&  & {\leftcat {b'}} \Delta  &  \xrightarrow[\textstyle \tau']{\kern2em}  &  1_\calC  & &  &  &  &  \nearrow \rlap{{\tau'}_c}  & \rlap{{\tau'}_\gamma} & \nearrow \rlap{{\tau'}_{c'}}  \\  &&&& &&&&&&   & {\leftcat b'}        \\  \\    \hline    &&&& &&&&&&  \kern6em & {\leftadj\bot}   \\  {\leftadj\bot}[\bigcirc] &&&&  & {\leftadj\bot} \Delta  &  \xrightarrow[\kern2em]{\textstyle \bigcirc}  &  1_\calC & &  &  &  &  \searrow \rlap{(\bigcirc_{\leftadj\bot} {=} 1_{\leftadj\bot})}  & & \kern2em \searrow \rlap{\bigcirc_{c}}    \\   \llap{ \text{(a special case)} \kern2em \leftcat\beta} \Bigg\downarrow &&&&  &  \llap{\leftcat\beta  \Delta} \Bigg \downarrow  &  [\calC, \calC]  &  \Bigg\Vert  & &  &  &  \llap{\leftcat \beta} \Bigg \downarrow  &  \calC  &  \leftadj\bot  &  \xrightarrow[\kern3em]{\gamma}  & c   \\   {\leftcat{b}}[\tau] &&&&  & {\leftcat {b}} \Delta  &  \xrightarrow[\textstyle \tau]{\kern2em}  &  1_\calC  & &  &  &  &  \nearrow \rlap{{\tau}_{\leftadj\bot}}  & {} \rlap{\tau_\gamma} & \nearrow \rlap{{\tau}_{c}}  \\  &&&& &&&&&&   & {\leftcat b}     \end{array}$$

Let $\boxed{ \leftcat b[\tau] = \leftcat b[ \leftcat b\Delta \buildrel \textstyle \tau \over \Rightarrow 1_{\calC} ] =  <\leftcat b, \leftcat b\Delta \buildrel \textstyle \tau \over \Rightarrow 1_{\calC}> }$ be an arbitrary cone in $(\Delta \downarrow \ulcorner 1_\calC \urcorner)$.

Since ($\tau$ is a cone), we have, for each $c \in \calC$,

$$\boxed{   \begin{array}{ccccc|c|ccccc} && \leftcat b &&     &&      \leftcat  b  & {} \rlap{ \kern-1em \xrightarrow[\kern9em]{\textstyle \tau_{\leftadj\bot}} }  &&& {\leftadj\bot} \\  & \llap{\tau_{\leftadj\bot}} \swarrow & \tau_{\bigcirc_c} & \searrow \rlap{\tau_c} &&  \text{i.e., reflecting,} &  &  \llap{\tau_c} \searrow & \tau_{\bigcirc_c} & \swarrow \rlap{ \bigcirc_c }   \\  \leftadj\bot & {} \rlap{\kern-1em \xrightarrow[\textstyle \bigcirc_c]{\kern9em}} &&& c & \kern6em &    && c   \end{array}    }$$

Thus $\boxed{ \tau_{\leftadj\bot} : \leftcat b \to {\leftadj\bot} }$ is an arrow in the comma category $(\Delta \downarrow \ulcorner 1_\calC \urcorner)$, $\tau_{\leftadj\bot}: \leftcat b[\tau] \to {\leftadj\bot}[\bigcirc]$.

It remains to show it is the unique such arrow.

Suppose $\boxed{ g : \leftcat b \to {\leftadj\bot} }$ is another such arrow in $(\Delta \downarrow \ulcorner 1_\calC \urcorner)$, $g : \leftcat b[\tau] \to {\leftadj\bot}[\bigcirc]$.

Then we have, for each $c \in {\calC}$ the commutative triangle above the single horizontal line, 

while specializing $c$ to be $\leftadj\bot$ gives the triangle below it:

$$\boxed{    \begin{array}{l|ccccc|c|cc}    &&&  {} \rlap{  \kern-4em \text{showing}  }   &&&&& & {} \rlap{ \kern-4em \text{showing}  }      \\     &&&  {} \rlap{  \kern-4em \text{(an initial ${\leftadj\bot} [\bigcirc]$ in $\calC$)}  }    &&&&&&  {} \rlap{  \kern-9em \text{(a limit $\rightadj{ \lim[\pi : {\lim}\leftadj\Delta \Rightarrow \rightcat{1_\calC}] }  \in \rightcat{ (\leftadj\Delta \downarrow \ulcorner 1_\calC \urcorner) }$ of $\rightcat{1_\calC}$)}  }       \\      &&&  {} \rlap{  \kern-4em \text{is (a limit of $\rightcat{1_\calC}$)}  }    &&&&& & {} \rlap{  \kern-4em \text{is (initial in $\calC$)}  }      \\ \hline  \\ \hline    \text{existence}  & \leftcat b & {}  \leftcat{    \rlap{   \kern-1em \xrightarrow [\kern11em]{  \textstyle \boxed{ \exists \; {\rightcat\tau}_{\leftadj\bot} }  }   }    }   &&& \leftadj\bot & \kern6em & &&     \\  && \rightcat{ \llap{\tau_c} \searrow }   &   {\rightcat\tau}_{\bigcirc_{\rightcat c}} & \swarrow \rlap{ \bigcirc_{\rightcat c} } && &   \rightadj\lim & {} \rlap{   \kern-1em \rightadj{  \xrightarrow [\kern9em] { \textstyle \smash{  \boxed{\exists \; \pi_{\rightcat c}} }  }  }   }  &&& \rightcat{c \, \forall}    \\   &&& \rightcat c && &&        \\   \hline  \text{uniqueness}   \\ \hline      \text{suppose} & \leftcat b[\tau] & {} \rlap{ \kern-1.5em \xrightarrow[\textstyle \kern4em g \kern4em]{\textstyle (\Delta \downarrow 1_\calC)} } &&&  {\leftadj\bot}[\bigcirc]   &&&&&   \\ \hline  \\ \hline   &  \leftcat b & {}\rlap{ \kern-1em \xrightarrow [\kern11em]{\textstyle g} }   &&& \leftadj\bot     &  \kern6em  &  && \rightadj\lim     \\   \text{then} && \llap{\tau_{\rightcat c}} \searrow & \rightcat{ \forall c }& \swarrow \rlap{ \bigcirc_{\rightcat c} }  && && \rightadj{ \llap{\pi_\lim} \swarrow }   &  \rightadj{  \pi_{ \pi_{\rightcat c} }  }  & \rightadj{ \searrow \rlap{\pi_{\rightcat c}} }   \\    &&&  \rightcat c    &&   &&     \rightadj\lim  & {} \rlap{  \kern-1em \rightadj{ \xrightarrow[\textstyle \pi_{\rightcat c}]{\kern9em} }  }  &&&  \rightcat{c \, \forall}    \\    \hline  \text{thus}  & \leftcat b & {} \rlap{ \kern-1em \xrightarrow[\kern11em]{\textstyle g} } &&& \leftadj\bot    &   &  && \rightadj\lim    \\  \text{in}   && \llap{\tau_{\leftadj\bot}} \searrow & \rightcat{ c = \leftadj\bot } & \swarrow \rlap{ \bigcirc_{\leftadj\bot} = 1_{\leftadj\bot} }      && &&  \rightadj{ \llap{ 1_{\rightadj\lim} = \pi_\lim } \swarrow  } & {\rightadj\pi}_f &   \rightadj{ \searrow \rlap{ \pi_{\rightcat c} }  }  \\  \text{particular}  &&& \leftadj\bot     &&   &&    \rightadj\lim  & {} \rlap{ \kern-1em \xrightarrow[\textstyle f \, \forall]{\kern9em} }  &&&  \rightcat{c \, \forall}  \end{array}    }$$      

But $\bigcirc_{\leftadj\bot} = 1_{\leftadj\bot}$.

Thus $g = \tau_{\leftadj\bot}$. 

Thus  (${\leftadj\bot} [\bigcirc]$ is terminal in $(\Delta \downarrow \ulcorner 1_\calC \urcorner)$, 

and thereby a limit of $1_{\calC}$. QED.

<hr />!

The following is the somewhat complicated version of the above used in the Adjoint Functor Theorem to prove that a limit in a certain comma category is initial, and thus constitutes a Left Adjoint.

$$\boxed{ \begin{array}{l|ccccc|c|cc} &&& {} \rlap{ \kern-4em \text{showing} } &&&&& & {} \rlap{ \kern-5em \text{showing} }   \\   &&& {} \rlap{ \kern-4em \text{(an initial ${\leftadj\bot} [\bigcirc]$ in $\calC$)} } &&&&&& {} \rlap{ \kern-14em \text{(a limit $\rightadj{ \lim[\pi : {\lim}\leftadj\Delta \Rightarrow \rightcat{1_{ \leftcat{(l \downarrow \rightadj R)} }}] } \in \rightcat{ (\leftadj\Delta \downarrow \ulcorner 1_{ \leftcat{(l \downarrow \rightadj R)} } \urcorner) }$ of $1_{ \leftcat{(l \downarrow \rightadj R)} }$)} }    \\    &&& {} \rlap{ \kern-4em \text{is (a limit of $\rightcat{1_\calC}$)} } &&&&& & {} \rlap{ \kern-7em \text{is (initial in $\leftcat{(l \downarrow \rightadj R)}$)} }   \\       \hline \\ \hline     \text{existence} & \leftcat b & {} \leftcat{ \rlap{ \kern-1em \xrightarrow [\kern11em]{ \textstyle \boxed{ \exists \; {\rightcat\tau}_{\leftadj\bot} } } } } &&& \leftadj\bot & \kern6em & &&   \\   && \rightcat{ \llap{\tau_c} \searrow } & {\rightcat\tau}_{\bigcirc_{\rightcat c}} & \swarrow \rlap{ \bigcirc_{\rightcat c} } && & {  \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] }  }   & {} \rlap{     \kern-1em \rightadj{ \xrightarrow [\kern18em] {      \textstyle \smash{  \boxed{ \exists \; \pi_{ \rightcat r \leftcat{[\kappa]} } }  }   }    }     } &&& \rightcat{ r \leftcat{[\kappa]} \, \forall } \\ &&& \rightcat c && && \\ \hline \text{uniqueness} \\ \hline \text{suppose} & \leftcat b[\tau] & {} \rlap{ \kern-1.5em \xrightarrow[\textstyle \kern4em g \kern4em]{\textstyle (\Delta \downarrow 1_\calC)} } &&& {\leftadj\bot}[\bigcirc] &&&&& \\ \hline \\ \hline & \leftcat b & {}\rlap{ \kern-1em \xrightarrow [\kern11em]{\textstyle g} } &&& \leftadj\bot & \kern6em & && \rightadj{  \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] }  }  \\ \text{then} && \llap{\tau_{\rightcat c}} \searrow & \rightcat{ \forall c }& \swarrow \rlap{ \bigcirc_{\rightcat c} } && && \rightadj{ \llap{\pi_{  \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] }  } } \swarrow } & \rightadj{ \pi_{ \pi_{ \rightcat r \leftcat{[\kappa]} } } } & \rightadj{   \searrow \rlap{  \pi_{ \rightcat r \leftcat{[\kappa]} }  }   }   \\   &&& \rightcat r \leftcat{[\kappa]}  && && \rightadj{  \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] }  }  & {} \rlap{ \kern-1em \rightadj{ \xrightarrow[\textstyle \pi_{ \rightcat r \leftcat{[\kappa]} }]{\kern18em} } } &&& \rightcat{ r \leftcat{[\kappa]} \, \forall } \\      \hline       \text{thus} & \leftcat b & {} \rlap{ \kern-1em \xrightarrow[\kern11em]{\textstyle g} } &&& \leftadj\bot & & && \rightadj{ \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] } }    \\    \text{in} && \llap{\tau_{\leftadj\bot}} \searrow & \rightcat{ c = \leftadj\bot } & \swarrow \rlap{ \bigcirc_{\leftadj\bot} = 1_{\leftadj\bot} } && && \rightadj{    \llap{   1_{  \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] }        } = \pi_{ \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] }  }   } \swarrow    }   & {\rightadj\pi}_f & \rightadj{   \searrow \rlap{   \pi_{ \rightcat r \leftcat{[\kappa]} }  }   } \\ \text{particular} &&& \leftadj\bot && &&   \rightadj{  \rightcat{ \big( (\black G Q) {\lim}_\calR \big) } \leftcat{ [\overline{\black G \lambda)}] }  } & {} \rlap{ \kern-1em \xrightarrow[\textstyle f \, \forall]{\kern18em} } &&& \rightcat{ r \leftcat{[\kappa]} \, \forall }  \\    \end{array} }$$      

<hr />

And here is a very general version of the argument, 

showing how (the situation) can perspicuously be viewed 2-categorically, 

as ( two (horizontal compositions) in (the 2-category $\CAT$) ).

(Writing $\rightadj{\boxed \lim}$ as short for $\rightadj{  \boxed{ \lim  \rightcat{1_\calC} }  }$.)

$$\boxed{     .\begin{array} {ccccccccc|c}   && \calI &&&& \calI &&   \\   & \llap{!} \nearrow &   \rightadj{ \llap{\pi} \swarrow \rlap{\kern-1.5em \swarrow} }  &  \rightadj{ \searrow \rlap{{\lim}} }  &&   \llap{!} \nearrow &   \rightadj{ \searrow \rlap{\kern-1.5em \searrow \kern0em \pi}  }  &  \rightadj{ \searrow \rlap{{\lim}} }  & &  \CAT   \\  \rightcat{   \calC \rlap{ \kern0em \xrightarrow[\textstyle 1_\calC]{\kern11em} }  }   &&&&  \rightcat{  \calC \rlap{ \kern0em \xrightarrow[\textstyle 1_\calC]{\kern11em} }   }    &&&& \rightcat\calC  \\   \hline  &&&&  \rightadj\lim  \\   &&& .\swarrow  &&  \rightadj{  \searrow \rlap{\lim \pi = \pi_\lim}  } &&     \\   &&  \rightadj{{\lim}}  &&  {} \rlap{  \kern-2em \rightadj{ \boxed{\pi \pi = \pi_\pi} }  }   &&  \rightadj{{\lim}}   &&&   \rightcat{[ \calC, \calC ]}     \\   &&&  \rightadj{  \llap{ \pi = \rightcat{1_\calC} \pi } \searrow   } &&  \rightadj{  \swarrow \rlap{ \pi \rightcat{1_\calC} = \pi }  }  \\  &&&& \rightcat{1_\calC}  \\         \hline  {} \rlap{  \kern1em  \text{Thus, by (the universal property)} }  \\  {} \rlap{ \kern1em \text{of (the $\rightadj{ \pi = \pi \rightcat{1_\calC} }$ at the lower right),} }  \\    {} \rlap{  \kern4em \boxed{  \rightadj{\pi_\lim = 1_\lim : \lim \to \lim} } \, . }   \\   {} \rlap{  \kern-1em \text{The important point here is the confluence of:} }  \\  {} \rlap{ \kern2em \text{the self-application (squaring) of} }  \\  {} \rlap{ \text{( (the $\rightadj\pi$ for $\rightadj{ \lim(\rightcat{1_\calC}) }$), an endo-2-cell on $\rightcat\calC$ ),} }  \\   {} \rlap{ \kern0em \text{and ( (the universal property) of (that $\rightadj\pi$) ).} }  \\   \hline     && \calI &&&& \calI &&   \\   &  \nearrow &  \rightcat{ \llap f \swarrow \rlap{\kern-1.5em \swarrow} }  &  \rightadj{ \searrow \rlap{{\lim}} }  &&   \llap{!} \nearrow &   \rightadj{ \searrow  \rlap{\kern-1.5em \searrow \kern0em \pi}    }  &  \rightadj{ \searrow \rlap{{\lim}} }  &&  \CAT   \\  \rightcat{   \calI \rlap{ \kern0em \xrightarrow[\textstyle c]{\kern11em} }  }   &&&&  \rightcat{  \calC \rlap{ \kern0em \xrightarrow[\textstyle 1_\calC]{\kern11em} }   }   &&&& \rightcat\calC  \\   \hline    &&&&  \rightadj\lim  \\   &&& \rightcat\swarrow  &&  \rightadj{  \searrow \rlap{\lim \pi = \pi_\lim}  }  &&     \\   &&  \rightadj{{\lim}}  &&  {} \rlap{   \kern-2em \rightcat{ \boxed{f \rightadj\pi = {\rightadj\pi}_f} }  }   &&  \rightadj{{\lim}}  &&&   \rightcat{ [ \calI, \calC ] \cong \calC }   \\   &&&  \rightadj{  \llap{  \pi_{\rightcat c} = \rightcat c \pi } \searrow   } &&  \rightcat{  \swarrow \rlap{ f \rightcat{1_\calC} = f }  }  \\  &&&& \rightcat c \\  \end{array}    }$$

Of course in each case what counts is 

the naturality of the right-hand occurrence of $\rightadj\pi$.