By definition $$f^{-1}f_{*}\mathcal{F}(U)=\lim_{\substack{\rightarrow\\V \supseteq f(U)}} \mathcal{F}(f^{-1}(V)).$$
If $V \supseteq f(U)$, then $U \subseteq f^{-1}(V)$. Since $\mathcal{F}$ is a sheaf, we have the restriction map: $$\operatorname{res}_{f^{-1}(V), U}:\ \mathcal{F}(f^{-1}(V)) \to \mathcal{F}(U).$$ My question is how could we obtain a natural map from $f^{-1}f_{*}\mathcal{F}(U)$ to $\mathcal{F}(U)$ by using the restriction maps? Thank you very much.
Expanding on Prahlad Vaidyanathan's comment:
Since $\mathcal{F}$ is a sheaf, for all open $W\supseteq V$ we have the restriction map $$\operatorname{res}_{f^{-1}(W),U}:\ \mathcal{F}(f^{-1}(W))\ \longrightarrow\ \mathcal{F}(U),$$ and since $\mathcal{F}$ is a sheaf, for all open $W_1\supseteq W_2\supseteq V$ $$\operatorname{res}_{f^{-1}(W_1),U}=\operatorname{res}_{f^{-1}(W_2),U}\circ\operatorname{res}_{f^{-1}(W_1),f^{-1}(W_2)}.$$ By the universal property of the direct limit, there exists a unique map $$\psi:\ \lim_{V\supseteq f(U)}\mathcal{F}(f^{-1}(V))\ \longrightarrow\ \mathcal{F}(U),$$ such that $\operatorname{res}_{f^{-1}(W),U}=\psi\circ\varphi_W$ for all open $W\supseteq U$, where $$\varphi_W:\ \mathcal{F}(f^{-1}(W))\ \longrightarrow\ \lim_{V\supseteq f(U)}\mathcal{F}(f^{-1}(V)),$$ is the canonical map.