Suppose I have a functor $f:(C,J)\to(D,K)$ between Grothendieck sites. Is there a condition on $f$ such that $f_!$ (the left adjoint to $f^*$) sends "$J$-epimorphisms", to $K$-epimorphisms, where by $J$-epimorphism I mean:
$h:X\to Y$ such that for all $C$, and all $y \in Y(C)$, there exists a cover $(g_i:C_i\to C)$ in $J$ and $y_i \in X(C_i)$ such that for all $i$, $Y(g_i)(y)=h(y_i)$.
EDIT: If X and Y are sheaves, then the notion of "J-epimorphism" coinincides with the categorical epis. As mentioned by David Brown, ANY left adjoint will preserves epis.
In fact, in the situation in which I was interested, I actually have such a (appropriate analogue of a) J-epimorphism between a sheaf and a stack, so, since f_! is a left adjoint, it will preserve this.