For a closed embedding (of varieties) $X\to Y$ let $U/X$ be etale. Is is true that there necessarily exists an etale $U'/Y$ such that $U'_X=U$? If this is wrong in general, are there any assumptions that make the statement correct ('my' $X$ is smooth, for example) or any ways to 'repair' it (in particular, I am interested in simplicial schemes).
I need this statement in order to prove my 'conjecture' Does one need to sheafify when defining the inverse image of a sheaf with respect to an embedding? The 'topological' case of my current question is trivial (for manifolds or metric spaces, at least), yet I do not understand what happens in the \'etale setting at all. Certainly, I would also be deeply grateful for any response to my previous question.:)
Upd. Now I also suspect that I need etale tubular neighbourhoods here. Yet any alternative suggestions could be very welcome!