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.

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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 $\rightadj\pi$.

A concrete introduction to adjunctions

DRAFT CURRENTLY UNDER REVISION!!!!

Let us start by considering two very concrete and visualizable categories, 

\[\begin{align}  &\rightcat{\mathcal R = \{ s < t \} \cong [1] = \bf 2 = \{ 0 < 1 \}}    \\    &\leftcat{\mathcal L = [3] = \bf 4 = \{ 0 < 1 < 2 < 3 \}}  \end{align}\]

These categories have familiar geometrical interpretations:

$\rightcat {\mathcal R = \{ s < t \}}$ as two points (vertices) connected by an arrow, i.e., a directed edge, 

$\leftcat{\mathcal L = [3] = \bf 4}$ as a tetrahedron whose vertices are $\leftcat{0,1,2,3}$, 

as in the right of the graphic below:

\[\begin{array}{cc|cccccc} & \kern1em & \kern1em & && \rightcat s \\ \rightcat {\mathcal R} &&& &&& \rightcat\searrow \\ &&&  &&&& \rightcat t \\ \rightadj{\Bigg\downarrow \rlap{R}} &&& &&& \\ &&& && \leftcat 1  &{}\rlap{\kern-1em \leftcat{\xrightarrow{\kern7em}}} &&& \leftcat 3 \\ \leftcat {\mathcal L} &&& & \leftcat\nearrow && \leftcat\searrow && \leftcat\nearrow \\ &&&  \leftcat 0 & {}\rlap{\kern-1em \leftcat{\xrightarrow{\kern7em}}} &&& \leftcat 2 \\ \end{array}\] 

Now consider the functor (which we shall denote as $\rightadj R$, at the left of the above graphic) 

from $\rightcat {\mathcal R}$ to $\leftcat {\mathcal L}$ 

which takes (the generic edge $\rightcat {\mathcal R}$) to (the central edge $\leftcat{(12)}$) in (the tetrahedron $\leftcat {\mathcal L}$), as shown at the right.

The question now is: 

Does (this specific functor $\rightadj R$) have (a left adjoint)?

The answer is:

No, not (a TOTAL left adjoint), but it does have (a PARTIAL left adjoint).

So, a better question in this situation is:

For which $\leftcat{ ( l = 0, 1, 2, \text{ or } 3 \in \mathcal L ) }$ does (a left adjoint $\leftcat l \leftadj L$) exist, and when it does, what is it?

There are three equivalent conditions that may be used in checking for this condition:

1. $\rightadj R$ has a (left adjoint, unit) pair $\leftcat l \leftadj L [\leftcat l \leftadj\eta]$, i.e. $\leftadj{   \langle (\leftcat l \xrightarrow{\; \leftcat l \leftadj\eta \;} \leftcat l \leftadj L \rightadj R), \leftcat l \leftadj L \rangle   }$. at $\leftcat{ l \in \mathcal L}$ 

2. The comma category $\leftcat {  ( l \downarrow {\rightadj R} ) } $ has an initial object $\leftcat l \leftadj L [\leftcat l \leftadj\eta]$.

3. The covariant functor $\big(  \leftcat{ \hom l {\mathcal L} {\rightadj R} } : \rightcat{\mathcal R \to {}} \Set (\text{or} \bftwo)  \big)$ is representable, with representation (universal element or universal arrow) $\leftcat l \leftadj L [\leftcat l \leftadj\eta]$.

When suitable limits exist (as they certainly do in the case of a finite linear order) and (a functor $\rightadj R$ preserves them), 

there is a formula for computing pointwise (the left adjoint $\leftadj L$) to (the functor $\rightadj R$)

(see, e.g., CWM Thm. X.1.2).

Given an object $\leftcat l$ in $\leftcat {\mathcal L}$, the formula is

\[  \leftcat l \leftadj L = \rightadj{ \lim \left[ \begin{matrix}{} \leftcat{ ( l \downarrow \rightadj R) } & \rightcat{\xrightarrow[\kern1em]{\textstyle Q = r}} & \rightcat{\mathcal R}   \\  \leftcat{ \langle (l \xrightarrow{\textstyle \lambda} \rightcat r  \rightadj R), \rightcat r \rangle = \rightcat r [\lambda] } & \rightcat\mapsto & \rightcat r \\ \end{matrix} \right] } , \]

where ($\rightadj\lim$, short for limit), is (the generalization to categories) of (the order-theoretic notion of infimum or greatest lower bound).

The situation is depicted in the following diagrams, 

where (the overline) denotes <i>both</i> 

(the projection map coming out of the limit) which corresponds to (an object $\leftcat{ \langle (l \xrightarrow[\lambda]{} \rightcat r  \rightadj R), \rightcat r \rangle = \rightcat r [\lambda] }$ in the comma category), AND

(the transpose under the adjunction) of (that object)!

\[ \begin{array} {cccc|c|cc} & \leftcat{\mathcal L} &&& \rightadj{ \xleftarrow[\kern3em]{\textstyle R} } & \rightcat{\mathcal R}   \\   \hline   \\   \leftcat l & \leftadj { \xrightarrow[\kern2em]{\textstyle \leftcat l \eta} } & \rightcat Q \rightadj\lim \rightadj R  & \kern3em && \rightcat Q \rightadj\lim   \\    & \leftcat { \llap\lambda \searrow } &  \rightadj{ \Bigg\downarrow  \rlap{ \Big( \overline{ \rightcat r \leftcat{[\lambda]} } \Big) R } }   &&& \rightadj{ \Bigg\downarrow  \rlap{  \overline{ \rightcat r \leftcat{[\lambda]} }  = \rightcat\pi_{ (\rightcat r \leftcat{[\lambda])} } } }  \\    && \rightcat r \rightadj R  &&& \rightcat{ r \rlap{ {} = ( \rightcat r \leftcat{[\lambda]} )Q } }  \end{array} \]

The above shows that $\rightcat Q\rightadj\lim [\leftcat l \leftadj\eta]$ is weakly initial in $\leftcat {  ( l \downarrow {\rightadj R} ) }$.

To prove $\rightcat Q\rightadj\lim [\leftcat l \leftadj\eta]$ initial we must show that $\overline{ \rightcat r \leftcat{[\lambda]} }$ is the unique arrow which makes the above commute.

That is a two-step process.

First we consider a special case, showing that $\overline{\rightcat Q\rightadj\lim[\leftcat l \leftadj\eta]}$ is $1_{\rightcat Q\rightadj\lim}$.

To compute these limits in the current case, see the following table:

\[\begin{array}{c|c|cc|c|cc|c} \leftcat {l \in \mathcal L} & \leftcat{(l \downarrow \rightadj R)} & \rlap{\kern-2em \leftcat{\text{arrows in left cat $\mathcal L$}}} && \rightcat{\text{$\rightadj\lim$ in $\mathcal R$}} & \rlap{\kern-.2em \rightcat{\text{arrows in right cat $\mathcal R$}}} \kern5em && \text{unit $\leftadj\iota : \leftcat l \mathrel{\leftadj\to} \leftcat l\leftadj L \rightadj R$}   \\  \hline   \leftcat 0 & \leftcat{(0 \downarrow \rightadj R)} & \leftcat { 0 \to {} } \rightcat s \rightadj R,   & \leftcat { 0 \to {} } \rightcat t \rightadj R & \rightcat s = \leftcat 0 \leftadj L & \rightcat{ s \to s}, & \rightcat{s \to t} & \leftadj\iota : \leftcat 0 \mathrel{\leftadj\to} \leftcat 0 \leftadj L \rightadj R = \rightcat s\rightadj R =  \leftcat 1   \\   \leftcat 1 & \leftcat{(1 \downarrow \rightadj R)}  & \leftcat 1 \mathrel{\leftcat\to} \rightcat s\rightadj R,  & \leftcat 1 \mathrel{\leftcat\to} \rightcat t \rightadj R & \rightcat s = \leftcat 1 \leftadj L & \rightcat{ s \to s}, & \rightcat{s \to t}  & \leftadj\iota : \leftcat 1 \mathrel{\leftadj\to} \leftcat 1\leftadj L\rightadj R = \rightcat s \rightadj R =  \leftcat 1   \\   \leftcat 2 & \leftcat{( 2 \downarrow \rightadj R)} && \leftcat 2 \mathrel{\leftcat\to} \rightcat t \rightadj R & \rightcat t = \leftcat 2 \leftadj L & & \rightcat{t \to t}  & \leftadj\iota : \leftcat 2 \mathrel{\leftadj\to} \leftcat 2\leftadj L\rightadj R = \rightcat t \rightadj R =  \leftcat 2   \\   \leftcat 3 & \leftcat{( 3 \downarrow \rightadj R)} &  \emptyset && \rightcat t & \emptyset \\ \end{array} \]

Two points worth noting:

1) (The arrows in (the right category) ) are (the projection arrows out of the limit) which correspond to (the arrows out of $\leftcat l$ in (the left category) ).

This gives one direction of the bijection between homsets of an adjunction.

2) Note that (the limit exists) even when ($\leftcat {l=3}$). In this case $\leftcat{ ( (l=3) \downarrow \rightadj R) }$ is $\emptyset$, the empty category, and (   the limit of ( the functor from (the empty category) into (another category) )   ) is always ( (a terminal object) in (that category) ), if either exists.

So $\rightadj{ \lim [\leftcat{ ( 3 \downarrow \rightadj R) } \rightcat{ {} \to {\mathcal R} }] }$ exists and is (the terminal object in $\rightcat {\mathcal R}$, namely $\rightcat t$). 

But there cannot be a homset bijection, because $\leftcat {\hom 3 {(\mathcal L = \bf 4)} {(\rightcat t \rightadj R = 2)} }$ is empty.

Alternatively, use the fact that (the functor $\leftcat{ \hom 3 {\mathcal L} {\rightadj R} }$) is (the empty functor), thus is not representable as a $\rightcat{ \hom r {\mathcal R} - }$, i.e., ($\leftcat 3 \leftadj L$ cannot exist).

<hr>

$$\begin{array}{lc|ccccc|cc} \rlap{\text{linearly-ordered sets,}} && \rlap{\text{diagram of objects and arrows in $\Cat$}}  &&&&& \rlap{\text{isomorphism of hom sets}}    \\     \rlap{\text{thus categories, i.e., objects in $\Cat$}} &&&&&&& \rlap{\text{(actually, of $\Set$-bimodules)}}    \\    \hline     \rightcat{\mathcal R = \{ s < t \} \cong [1] = \bf 2 = \{ 0 < 1 \}} &&&&& \rightcat{(\mathcal R \cong [1]=\bf 2)}  &&& \rightcat{ \hom {\leftadj L_{\leftcat -}} {\mathcal R}  - } \\    \leftcat{\mathcal B = [2] = \bf 3 = \{ 0 < 1 < 2 \}} &&&& \leftadj{\llap L \nearrow} & \rightadj{\big\downarrow \rlap R} &&& \wr\Vert   \\    \leftcat{\mathcal L =  [3] = \bf 4 = \{ 0 < 1 < 2 < 3 \}} & \kern1em & \kern1em & \leftcat{(\mathcal B=[2]=\bf 3)} & \leftcat{\xrightarrow[\textstyle J]{}} & \leftcat{(\mathcal L =[3]=\bf 4)}  & \kern1em & \kern1em & \leftcat { \hom {J_-} {\mathcal L} {\rightadj R_{\rightcat -}} } \end{array}$$

(Looking ahead, there is a notation for the situation in $\Cat$ described by the two right columns above: $\leftadj L \mathrel{\hom {\rightcat J} \dashv {}} \rightadj R$. 

It is called a relative adjunction: 

$L$ is left adjoint to $R$ relative to $J$. 

Note that if $\leftcat J$ had been the identity functor on $\leftcat {\mathcal L}$, we would have an ordinary adjunction between $\rightcat {\mathcal R}$ and $\leftcat {\mathcal L}$.)

and $\leftcat{\mathcal B = [2]}$ as the initial face of $\leftcat{\mathcal L = [3]}$.

These categories are related by a number of adjunctions, partial adjunctions, and relative adjunctions.

Because (these categories are so concrete and familiar), 

it is easier to understand the adjunctions relating (these specific categories) than it is to understand (the general case). 

To achieve some degree of specificity, concreteness, and hopefully clarity, in our further discussions, 

we are going to focus on (one specific example)

<hr>

\[ \begin{array}{}   \leftcat{(l \downarrow \rightadj R)}  & {} \rlap{ \rightcat{ \xrightarrow[\kern4em]{\textstyle Q} } }  &&& \rightcat{\mathcal R}    \\      &  \llap ! \searrow & \rightadj{ \llap\pi \Big\Uparrow }  & \rightadj{ \nearrow  \rlap{\kern-1.5em \rightcat Q \lim} }  &  \leftadj{ \Big\Uparrow \rlap \eta }  & \rightadj{ \searrow \rlap R }    \\   && \mathcal I  & {} \rlap{ \leftcat{ \xrightarrow[\textstyle l]{\kern4em} } }  & &&  \leftcat{ \mathcal L }  \\  \end{array} \]

<hr>

References:

https://ncatlab.org/nlab/show/relative+adjoint+functor