TITLE: Inverse image presheaf
QUESTION [5 upvotes]: Let $f:X\rightarrow Y$ be a continuous map of topological spaces, and $\mathscr{G}$ a sheaf on $Y$. So far I failed to come up with a simple example where the presheaf $f^{-1}\mathscr{G}$ on $X$ obtained via the direct limit
$$f^{-1}\mathscr{G}(U):=\lim_{f(U)\subset V}\mathscr{G}(V)$$
is not a sheaf. If anyone could give me an example (the simpler the better), it would be greatly appreciated.

REPLY [7 votes]: Take $X = $ two points with the discrete topology and $Y = $ a point. Let $\mathscr{G}$ be a set with more than two elements, sitting on $Y$. What you get is the corresponding constant pre-sheaf on $X$, which fails to satisfy the gluing axiom.