I think I need some help on a problem about sheaf theory.
Suppose that $f: X \rightarrow Y$ is a continuous map of topological spaces, $\mathscr{F}$ is a sheaf on $X$. Prove that the morphism of sheaves $\rho: f^{-1}f_* \mathscr{F} \rightarrow \mathscr{F}$ can be obtained from the morphism of presheaves $Pf^{-1}(f_*\mathscr{F}) \rightarrow \mathscr{F}$.
Here, $f_*$ is the direct image functor, and $f^{-1}$ is the inverse image functor. $Pf^{-1}$ and $f^{-1}$ are connected by sheafification:
Suppose that $f: X \rightarrow Y$ is a continuous map of topological spaces, $\mathscr{G}$ is a sheaf on $Y$. Then for any open subset $U \subset X$, define $(Pf^{-1} \mathscr{G}) (U) = \varinjlim_{V \supseteq f(U), V \in \mathfrak{O}(Y)} \mathscr{G}(V),$ then $\{ (Pf^{-1} \mathscr{G})(U) \}$, together with the restriction map, becomes a presheaf. The sheaf associated to $Pf^{-1}\mathscr{G}$ is called the inverse image functor, denoted $f^{-1}\mathscr{G}$.
There are quite a lot of concepts and definitions in sheaf theory. I am afraid I am about to mess them up in my head. As to this problem, although I can find the definition of all things on the book, I don't know what to do.
Would you please give me some help? Thanks in advance!