TITLE: Does one need to sheafify when defining the inverse image of a sheaf with respect to an embedding?
QUESTION [8 upvotes]: 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)?

REPLY [2 votes]: Perhaps I'm thinking of sheaves a bit differently from you here, but as long as we think of sheaves as \'{e}tale maps this holds for arbitrary continuous maps $f:X\to Y$ and not just open or closed embeddings.  The reason for this is that, \'{e}tale maps are stable under pullback along arbitrary continuous maps.  See, e.g., Mac Lane and Moerdijk (Sheaves in Geometry and Logic) Lemma 1 (II.9, page 99).  
I guess that you know this though and are asking about the the case where we are not regarding sheaves as \'{e}tale maps.  In this case you are right that some sheafification is required (since the construction of the inverse image along $f$ involves taking a colimit in $\text{Sh}(X)$). (I had thought that this was incorrect, but I had misunderstood precisely how you were claiming one should compute the inverse image in this case and so have removed my counterexample since it did not quite address your question.)