I am currently working with the pullback of sheaves of $\mathcal{O}$-modules. Let $f:Y\rightarrow X$ be a continuous map of topological spaces and let $\mathcal{F}$ be a sheaf on $X$. Then the inverse image sheaf $f^{-1}\mathcal{F}$ is the sheaf on $Y$ associated to the presheaf $V\mapsto \lim\mathcal{F}(W)$ where $V$ is any open set in $Y$ and the direct limit is taken over all open subsets $W$ of $X$ such that $f(V)\subset W$.
We define $f^{*}\mathcal{F}$ to be the sheaf associated to the tensor product presheaf $$V\mapsto (f^{-1}\mathcal{F})(V)\otimes_{(f^{-1}\mathcal{O}_{X})(V)}\mathcal{O}_{Y}(V).$$
Goal: Whenever $f(V)\subset U$ for opens $V\subset Y$ and $U\subset X$ I want to construct a map $f^{*}:\mathcal{F}(U)\rightarrow (f^{*}\mathcal{F})(V)$, such that in particular we have a map between the global sections.
Note that when $f(V)\subset U$ for opens $V\subset Y$ and $U\subset X$ we do have a map $$\mathcal{F}(U)\rightarrow \lim \mathcal{F}(W), s\mapsto [(U,s)],$$ where the limit is again take over all open subset $W\subset X$ containing $f(V)$.
Problem: How can I find a map from $\lim \mathcal{F}(W)$ into $(f^{-1}\mathcal{F})(V)$?
For a map from $(f^{-1}\mathcal{F})(V)\rightarrow (f^{-1}\mathcal{F})(V)\otimes_{(f^{-1}\mathcal{O}_{X})(V)}\mathcal{O}_{Y}(V)$ we can just send $r\mapsto r\otimes 1$.
Last problem: How does the map from $(f^{-1}\mathcal{F})(V)\otimes_{(f^{-1}\mathcal{O}_{X})(V)}\mathcal{O}_{Y}(V)$ to $f^{*}\mathcal{F}(V)$ look like?
As one might have noticed all the maps that we still need have to deal with the sheafification, and I think that this is where my main struggle is. Any help would be appreciated!