Assume we are given a map $f: X \rightarrow Y$ between topological spaces which is open, surjective and has (pathwise) connected fibers. Consider categories $\text{Sh}(X),\text{Sh}(Y)$ of sheaves (with values in any given fixed sensible category) on $X$ and $Y$. The inverse image functor $f^\ast: \text{Sh}(Y) \rightarrow \text{Sh}(X)$ will then be fully faithful and hence we can consider $\text{Sh}(Y)$ up to equivalence as a full subcategory of $\text{Sh}(X)$. Obviously a necessary condition for a sheaf to lie in this subcategory is that it is constant when restricted to the fibers of $f$.

My question is: Is this condition also sufficient? Is it even sufficient for trivial fibrations $F\times Y \rightarrow Y$  ?

Assume we are given a map $f: X \rightarrow Y$ between topological spaces which is open, surjective and has (pathwise) connected fibers. Consider categories $\text{Sh}(X)$, $\text{Sh}(Y)$ of sheaves (with values in any given fixed sensible category) on $X$ and $Y$. The inverse image functor $f^\ast: \text{Sh}(Y) \rightarrow \text{Sh}(X)$ will then be fully faithful and hence we can consider $\text{Sh}(Y)$ up to equivalence as a full subcategory of $\text{Sh}(X)$. Obviously a necessary condition for a sheaf to lie in this subcategory is that it is constant when restricted to the fibers of $f$.

My question is: Is this condition also sufficient? Is it even sufficient for trivial fibrations $F\times Y \rightarrow Y$?