Obtaining a morphism of sheaves from a morphism of presheaves 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!
 A: The main tool here is the universal property of sheafification, to wit: if $\mathcal E$ is a presheaf on $X$ and $\mathcal F$ a sheaf on $X$, there is a functorial (in $\mathcal E,\mathcal F$) bijection 
$$  Hom_{Presh_X}(\mathcal E,presh(\mathcal F))  \simeq    Hom_{Sh_X}(sh(\mathcal E),\mathcal F)\quad (*)$$ 
[If you know the terminology, you are in presence of a couple of adjoint functors: the important sheafification   functor and the trivial presheafification or forgetful functor]     
In order to  apply this to your situation, set $\mathcal E=Pf^{-1}(f_*\mathcal F)$ . So the displayed isomorphism  $(*)$ tells you that you have to exhibit a morphism      $Pf^{-1}(f_*\mathcal F) \to \mathcal F$  .
If you go back to your  definition with inductive limits of $Pf^{-1}(f_*\mathcal F)$, you'll have to define ( in the  notation of your second greyed box)   compatible maps $f_*\mathcal F(V)= \mathcal F(f^{-1}(V)) \to \mathcal F(U)$. And nothing could be easier: take restriction, since  $V\supseteq f(U)$ implies $ f^{-1}(V)\supseteq U$ !
