Since there is an isomorphism of functors $\colim \cong \pi_0 \Elts$ (Guitart & den Bril, 1977) [this means that the square is (pseudo)commutative, i.e. that the colimit of a functor $F : C \to \Set$ is the $\pi_0$ of its category of elements, TN] we can make condition 2 above explicit, and $\vf$ exact equals the following criterion ZZ:

• (ZZ1) for all $x\in X_0, y\in Y_0$, $p : Ux \to Vy$ in $B$, there exist $a\in A, \, m : x \to Sa, \, n : Ta \to y$ with $Vn\circ \vf_a\circ Um =p$;
• (ZZ2) For every $(a,m,n), (\bar a,\bar m, \bar n)$ such that $Vn\circ \vf_a\circ Um = V\bar n\circ \vf_{\bar a}\circ U\bar m$, there exists a zig-zag of maps [a headless dotted arrow denotes a morphism in either direction connecting $u,v\in A$, TN] in $A$ such that applying $T$ and $S$ as follows there exist ‘lantern’ diagrams as in the following picture: [i.e. with the property that the components of $\vf$ “connect” the upper half and the lower half of the “Chinese lantern” diagram on the right. TN]