This seems to be a rather simple (stupid?:)) question; yet I was not able to find an answer quickly.

For a morphism $f:X\to Y$ of schemes (or topological spaces) and an (etale or topological) sheaf $f$ (of abelian groups) on $Y$ one defines the inverse image $f^*(S)$ (or is $f^{-1}$ more standard?) as the sheafification of the inverse image of $S$ considered as a presheaf.

Now suppose that $f$ is a (closed) embedding. Is the sheafification necessary here? It seems that the answer is no, since any covering of an open $U\subset X$ could be presented as a 'limit' of coverings of open $V\subset Y$, $V\supset f(U)$ (in the topogical context); so the 'presheaf inverse image' of a sheaf is a sheaf also. This seems to be easy, as well as carrying this argument over to the etale context; yet I would be deeply grateful for a definite answer and a reference for it. Also, does the situation change when one passes to simplicial schemes and sheaves? Is this fact 'standard' (if it is true)?

1more comments