Beck-Chevalley condition on pushouts Let $C$ be a regular category with pushouts and $S(X)$ is the lattice of subjects of $X$. For every arrow $f\colon X\to A$, pulling back along $f$ gives a map $f^*\colon S(A)\to S(X)$ which has a left adjoint $f_*$. These maps satisfy the Beck-Chevalley condition on a square $fg=hk$ if $h^*f_*=k_*g^*$.
The BC condition holds on all pullbacks. Does it hold also on pushouts? If no, do you know a class of regular categories where this is true (exact, abelian, pretoposes...)?
 A: No, the Beck-Chevalley condition does not hold for all pushout squares in a regular category, and not even if the category is exact, or a pretopos, or even a topos.  In fact, here is a counterexample in $\rm Set$.  Let $B = \{a,b\}$ and $C = \{\alpha,\beta\}$ and $A=\{0,1,2\}$, define $f:A\to B$ by $f(0)=f(1)=a$ and $f(2)=b$, and $g:A\to C$ by $g(0)=\alpha$ and $g(1)=g(2)=\beta$.  Let $P$ be the pushout of $f$ and $g$ with coprojections $p:B\to P$ and $q:C\to P$.  Then
$$q(\alpha) = q(g(0)) = p(f(0)) = p(a) = p(f(1)) = q(g(1)) = q(\beta) = q(g(2)) = p(f(2)) = p(b)$$
so $P$ is a one-element set.  This, if we let $U=\{b\} \in S(B)$, then $p_!(U)$ is all of $P$, hence $q^* p_!(U)$ is all of $C$.  (The notation $p_!$ for left adjoints of $p^*$ is, I think, more common than $p_*$ in this field, the latter being used more often for right adjoints.)  But $f^*(U) = \{2\}$, so $g_! f^*(U) = \{\beta\} \neq q^* p_!(U)$.
There is something positive that can be said, however.  Suppose $P = B \amalg_A C$ is a pushout of $f:A\to B$ and $g:A\to C$ such that the induced map $\Delta : A \to B\times_P C$ is a regular epimorphism.  Then the pullback square defining $B\times_P C$, consisting of $h:B\times_P C\to B$ and $k:B\times_P C\to C$ with $p:B\to P$ and $q:C\to P$, does satisfy the Beck-Chevalley condition.  Moreover, since $\Delta$ is a regular epimorphism, we have $\Delta_! \Delta^* = \rm Id$, and therefore
$$ g_! f^* = k_! \Delta_! \Delta^* h^* = k_! h^* = q^* p_! $$
so the original pushout square does satisfy the Beck-Chevalley condition.
The above "zig-zag" counterexample is simply the "minimal" situation in which $\Delta$ fails to be surjective.  (One might call it "the minimal way to violate the hypotheses of the baby Blakers-Massey theorem".)  Thus, it seems that $\Delta$ being a regular epi is probably the best possible hypothesis.
