The question of proving the projection formula has been asked before on MSE but for spaces & sheaves with more structure. Here, I only want $\mathcal G, \mathcal F$ to be sheaves on $Y , X$ respectively (Hausdorff, connected, semilocally simply connected, among other things...), and $f: X \to Y$ some continuous morphism, and to show that $$ f_* (f^{-1} \mathcal F \otimes \mathcal G) \cong \mathcal F \otimes f_* \mathcal G.$$
I have already shown that $f_*$ distributes over $\otimes$, so all I need now is to show that $f_* f^{-1} \mathcal F \cong \mathcal F$. I know that $f_*$ and $f^{-1}$ are an adjoint pair, but to me, all that means is that $\text{Hom}_X(f^{-1} \mathcal F, \mathcal G) \cong \text{Hom}_Y(\mathcal F, f_* \mathcal G)$. I know the result that if $\phi : f_* f^{-1} \mathcal F \to \mathcal F$ is a sheaf morphism whose induced morphisms on the stalks are isomorphisms, then $\phi$ is also an isomorphism, but I am struggling to concretely write down the stalks of $f_* f^{-1} \mathcal F$. I have also tried using the universal property of the pull-back sheaf $f^{-1} \mathcal F$ but I don't think that goes anywhere.
Edit: As Roland points out, this is not correct in general, but I have sketched out a 'proof' and am not sure where it fails, even when working with the counterexample. Let $\otimes_{ps}$ denote the tensor product of presheaves, and first consider $\mathcal F, \mathcal G$ as presheaves.
I get $$ f_* (f^{-1} \mathcal F \otimes_{ps} \mathcal G)(U) = (f_* f^{-1} \mathcal F \otimes_{ps} f_* \mathcal G )(U),$$ since I am quite sure that $f_*$ distributes over $\otimes_{ps}$ (so I think by the universal property of sheafification this would also hold over their sheafifications, which are isomorphic to $\mathcal F, \mathcal G$ respectively?)
Then by definition of $\otimes_{ps}$ this equals $$ f_*f^{-1} \mathcal{F} (U) \otimes f_* \mathcal{G} (U),$$ so all I need now is to show that $ f_*f^{-1} \mathcal{F} (U) = \mathcal{F} (U)$. But this is not so hard, if I work with the presheaf associated to the pullback sheaf $f^{-1} \mathcal F$: applied here, I get $$ (\text{ps}f^{-1}\mathcal F )(f^{-1}(U)) = \{(V,s) : V \supset U, s \in \mathcal{F}(V)\}/\sim $$ where $\sim$ is the quotient relation that we get from the definition of stalks.
Now assuming this is the case, any $(V,s)$ is equivalent to $(U, s\vert_U)$, since the restriction of $s$ to $U = U \cap V$ is exactly $s\vert_U$. Thus there is a morphism of sections, which commutes with the restriction map.
Where could I have gone wrong here?