I'm in need of a condition that is analogous to the "finality" condition in the following lemma:


Lemma: A functor $F\colon A\to B$ is final if and only if for any functor $x\colon B\to Set$, the natural map $colim (xF)\to colim(x)$ is an isomorphism.


This lemma could be taken instead as a definition of *final functor*, but finality is more easily recognized by whether all slice categories of a certain kind are non-empty and connected. I want a recognition principle for a more general kind of finality, which I'm calling local finality.


The more general context requires a bit of notation. If $A$ is a category, write $A-Set$ for the category of functors $A\to Set$. If $F\colon A\to B$ is a functor, write $\Delta\_F\colon B-Set\to A-Set$ for the ``composition with $F$" functor, and write $\Sigma\_F$ for its left adjoint and $\Pi\_F$ for its right adjoint (these three are also sometimes denoted by $F^\*, F\_!$, and $F\_\*$ respectively).


The following lemma (obviously) holds for some appropriate definition of *locally final*. 


Lemma: Suppose that we have a commutative diagram $A\xrightarrow{F}B\xrightarrow{x}C$ and let $G:=xF$. Then $F$ is locally final if and only if the natural map $\Sigma\_G\Delta\_F\to\Sigma\_x$ is an isomorphism.


Is there a nice recognition principle for this kind of ``local finality"? I have a big messy condition obtained by following my nose, but it's of no use. Any help would be greatly appreciated.


Thanks!



Local finality condition (for re-indexing parameterized colimits)
Since nobody has said so, I will mention that the notion you describe is a particular case of the known --- but perhaps obscure --- concept of Guitart exact square. One can read about it in [the nlab page](http://ncatlab.org/nlab/show/exact+square) and in [an article by Maltsiniotis](http://arxiv.org/abs/1101.4144). Even though the latter article aims to generalize exact squares to a homotopical context, it still gives a good, if somewhat skewed, overview of the concept.


To justify the relative usefulness of exact squares, let me state that instances of that notion characterize: fully faithful functors, (co)final functors, initial functors, absolute Kan extensions, among other concepts (including my personal favourite, [absolutely dense functors](http://www.tac.mta.ca/tac/volumes/8/n20/8-20abs.html)).


For completeness, I will summarize some characterizations of the notion of exact square. A square of small categories, functors, and natural transformations:
$$ \begin{matrix} 
A & \overset{U}{\longrightarrow} & B \\
\llap{\scriptstyle L}\Big\downarrow & \big\Downarrow\rlap{\scriptstyle\alpha} & \Big\downarrow\rlap{\scriptstyle R} \\
A' & \underset{D}{\longrightarrow} & B'
\end{matrix} $$
(i.e. a natural transformation $\alpha:R\circ U\to D\circ L$) is called *exact* if any of the following equivalent conditions hold:


1. The natural 2-cell induced by $\alpha$ in
$$ \begin{matrix} 
\textrm{Set}^A & \overset{\Delta\_U}{\longleftarrow} & \textrm{Set}^B \\
\llap{\scriptstyle \Sigma\_L}\Big\downarrow & \big\Downarrow & \Big\downarrow\rlap{\scriptstyle \Sigma\_R} \\
\textrm{Set}^{\smash{A'}} & \underset{\Delta\_D}{\longleftarrow} & \textrm{Set}^{\smash{B'}}
\end{matrix} $$
is an isomorphism. That is, the induced natural transformation $\Sigma\_L\circ\Delta\_U \to \Delta\_D\circ\Sigma\_R$ is an isomorphism.
2. For every $x\in A'$, the naturally induced functor on over-categories
$$ A/x=L/x \longrightarrow R/(D(x))=B/(D(x)) $$
is (co)final.
3. For every $x\in B$, the naturally induced functor on under-categories
$$ x/A=x/U \longrightarrow (R(x))/D=(R(x))/A' $$
is initial.
4. For each object $x\in B$ and $y\in A'$, and each arrow $f:R(x) \to D(y)$ in $B'$, the category of factorizations $C\_{x,y,f}$ is connected (which I, Karol, and [a surprisingly large/vocal set of mathematicians](https://mathoverflow.net/questions/120536/is-the-empty-graph-a-tree) take to mean non-empty). Here, the category $C\_{x,y,f}$ is defined by:
	* the objects of $C\_{x,y,f}$ are triples $(z,g,h)$ where $z$ is an object of $A$, $g:x\to U(z)$ is a morphism in $B$, and $h:L(z)\to y$ is a morphism of $A'$, such that $D(h)\circ\alpha\_z \circ R(g)=f$;
	* a morphism $(z,g,h)\to (z',g',h')$ in $C\_{x,y,f}$ is an arrow $k:z\to z'$ such that $g'= U(k)\circ g$ and $h=h'\circ L(k)$.
5. For all objects $x\in B$ and $y\in A'$, the natural map from the coend
$$ \int^{a\in A} B(x,U(a))\times A'(L(a),y) \longrightarrow B'(R(x),D(y)) $$

is an isomorphism of sets.



Before proceeding, observe that condition 4 is simply a restatement of conditions 2 and 3. In fact, the categories of factorizations $C\_{x,y,f}$ defined in 4 are exactly the categories whose connectedness must be checked to ensure that the functor in condition 2 is cofinal (respectively, that the functor in condition 3 is initial). More precisely, the categories $C\_{x,y,f}$ are the under-categories $a/F$ of the functor $F$ in condition 2 (respectively, the over-categories of the functor in condition 3) for objects $a$ in the codomain of $F$. I feel this both motivates and gives a nice way to remember the definition of $C\_{x,y,f}$.


Then $A\overset{F}{\rightarrow}B\overset{x}{\rightarrow}C$ is locally final (in the sense David Spivak states) if and only if the the square
$$ \begin{matrix} 
A & \overset{F}{\longrightarrow} & B \\
\llap{\scriptstyle G}\Big\downarrow & \big\Downarrow\rlap{\scriptstyle\textrm{id}\_G}\ & \Big\downarrow\rlap{\scriptstyle x} \\
C & \underset{\textrm{id}\_C}{\longrightarrow} & C
\end{matrix} $$
(filled by the the identity 2-cell on $G=x\circ F$) is exact. Under this interpretation, condition 4 above is exactly the condition given in Karol Szumiło's answer.


**Addendum:** To finish off, here are a few further equivalent characterizations of the exactness of the original square ($\alpha:R\circ U\to D\circ L$) drawn at the top of this answer:


* For any cocomplete category X, the natural 2-cell induced by $\alpha$ in
$$ \begin{matrix} 
X^A & \overset{\Delta\_U}{\longleftarrow} & X^B \\
\llap{\scriptstyle \Sigma\_L}\Big\downarrow & \big\Downarrow & \Big\downarrow\rlap{\scriptstyle \Sigma\_R} \\
X^{\smash{A'}} & \underset{\Delta\_D}{\longleftarrow} & X^{\smash{B'}}
\end{matrix} $$
is an isomorphism. Note that for $X=\textrm{Set}$, we recover condition 1 above.
* The natural 2-cell induced by $\alpha$ in
$$ \begin{matrix} 
\textrm{Set}^A & \overset{\Pi\_U}{\longrightarrow} & \textrm{Set}^B \\
\llap{\scriptstyle \Delta\_L}\Big\uparrow & \big\Uparrow & \Big\uparrow\rlap{\scriptstyle \Delta\_R} \\
\textrm{Set}^{\smash{A'}} & \underset{\Pi\_D}{\longrightarrow} & \textrm{Set}^{\smash{B'}}
\end{matrix} $$
is an isomorphism.
* For any complete category X, the natural 2-cell induced by $\alpha$ in
$$ \begin{matrix} 
X^A & \overset{\Pi\_U}{\longrightarrow} & X^B \\
\llap{\scriptstyle \Delta\_L}\Big\uparrow & \big\Uparrow & \Big\uparrow\rlap{\scriptstyle \Delta\_R} \\
X^{\smash{A'}} & \underset{\Pi\_D}{\longrightarrow} & X^{\smash{B'}}
\end{matrix} $$
is an isomorphism. Note that for $X=\textrm{Set}$, we recover the preceding condition.
* The opposite square
$$ \begin{matrix} 
A^{\textrm{op}} & \overset{L^{\textrm{op}}}{\longrightarrow} & {A'}^{\textrm{op}} \\
\llap{\scriptstyle U^{\textrm{op}}}\Big\downarrow & \big\Downarrow\rlap{\scriptstyle\alpha^{\textrm{op}}} & \Big\downarrow\rlap{\scriptstyle D^{\textrm{op}}} \\
B^{\smash{\textrm{op}}} & \underset{R^{\textrm{op}}}{\longrightarrow} & {B'}^{\smash{\textrm{op}}}
\end{matrix} $$
is exact. Note that the preceding 3 conditions and condition 1 applied to this opposite square give diagrams with categories of presheaves (contravariant functors) on $A$, $B$, $A'$, and $B'$, instead of categories of covariant functors on those categories. In fact, a common (equivalent) definition of exactness is the analog of condition 1 for presheaves: the 2-cell in the square
$$ \begin{matrix} 
\widehat{A} & \overset{\Sigma\_{U^{\textrm{op}}}}{\longrightarrow} & \widehat{B} \\
\llap{\scriptstyle \Delta\_{L^{\textrm{op}}}}\Big\uparrow & \big\Downarrow\rlap{\scriptstyle} & \Big\uparrow\rlap{\scriptstyle \Delta\_{R^{\textrm{op}}}} \\
\widehat{A'} & \underset{\Sigma\_{D^{\textrm{op}}}}{\longrightarrow} & \widehat{B'}
\end{matrix} $$
is an isomorphism.


