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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.

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Suppose I have a functor $f:(C,J)->(D,K)$ 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$y \in$Y(C)$, Y(C)$, there exists a cover $(g_i:C_i\to C)$ in $J$ and $y_i$ y_i \in $X(C_i)$ such that for all $i$, $Y(g_i)(y)=h(y_i)$.

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Suppose I have a functor f:(C,J)->(D,K) $f:(C,J)->(D,K)$ between Grothendieck sites. Is there a condition on f $f$ such that f_! $f_!$ (the left adjoint to f^*) $f^*$) sends "J-epimorphisms", $J$-epimorphisms", to K-epimorphisms, $K$-epimorphisms, where by J-epimorphism $J$-epimorphism I mean:

h:X->Y

$h:X\to Y$ such that for all C, $C$, and all y $y$ in Y(C), $Y(C)$, there exists a cover (g_i:C_i->C) $(g_i:C_i\to C)$ in J $J$ and y_i $y_i$ in X(C_i) $X(C_i)$ such that for all i Y(g_i)(y)=h(y_i).$i$, $Y(g_i)(y)=h(y_i)$.

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