I'm trying to show that the tensor product of sheaves commutes with inverse image. I've reduced the problem to the following isomorphism
$$f_*\mathscr{H}om_X(f^*\mathcal{N},\mathcal{P}) \cong \mathscr{H}om_Y(\mathcal{N},f_*\mathcal{P})$$
where $ f:X\rightarrow Y$ is a morphism of ringed spaces, $\mathcal{N}$ is a $\mathcal{O}_Y$ module, and $\mathcal{P} $ is a $ \mathcal{O}_X$ module.
I'm trying to prove this via the adjunction of $f^* $ and $f_ *$, but I'm unable to. Can someone guide me through the steps involved in constructing this isomorphism?
$\def\HH{{\mathcal{H}om}}\def\Hom{{\operatorname{Hom}}}\def\N{{\mathcal N}}\def\P{{\mathcal P}}$This comes from checking the natural isomorphism from adjunction on each open set and using naturality to see it induces a morphism of sheaves not just sets.
Explicitly, take an open set $U \subset Y$. Then
$$ f_*\HH_X(f^*\N,\P)(U) = \Hom_{f^{-1}(U)}\left(f^*\N|_{f^{-1}(U)},\P|_{f^{-1}(U)}\right) \enspace \enspace \enspace (*) $$
but $f^*\N|_{f^{-1}(U)} = \left(f|_{f^{-1}(U)}\right)^*\N|_U$. Then by the adjunction, (*) is naturally isomorphic to
$$ \Hom_U\left(\N|_U, f_*\P|_U\right) = \HH_Y(\N,f_*\P)(U). $$
Naturality of this isomorphism tells us that this map commutes with the restrictions and so it is a map of sheaves and so we get an isomorphism.