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 and in an article by Maltsiniotis. 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).

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:

- 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.
- 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.
- 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.
- 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 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 $h=h'\circ L(k)$.

- 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.