In Arapura's Algebraic Geometry over the Complex Numbers, exercise 2.1.16. reads:
Let $F:X\to Y$ be a surjective continuous map. Suppose that $\mathscr{P}$ is a sheaf of $T$-valued functions on $X$. Define $f\in \mathscr{Q}(U) \subset \text{Map}_T(U)$ if and only if its pullback $F^{*}f=f\circ F|_{f^{-1}U}$ belongs to $\mathscr{P}(F^{-1}(U))$. Show that $\mathscr{Q}$ is a sheaf on $Y$.
($X,Y$ are topological spaces)
My question is with the notation $f\circ F|_{f^{-1}U}$. I can gather that $U$ is an open subset of $Y$, so since $f$ is a map from $U$ to $T$, I'm not sure what the restriction $F|_{f^{-1}U}$ is supposed to mean unless the author intended to write $F^{*}f=f\circ F|_{F^{-1}U}$. Does this seem to be what's going on, or am I missing something?
Yes, it's a typo. It should read $F^{-1}U$.