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
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