Disclaimer: A question about the same result has been asked here previously. See my comment at the bottom for why I think this question is unique.
Let $\phi: A \longrightarrow B$ be a morphism of rings with corresponding morphism of affine schemes, $f: \text{Spec }B \longrightarrow \text{Spec }A$. I will denote $Y = \text{Spec }B$ and $X = \text{Spec }A$. Let $N$ be a $B$-module and denote by $_A{N}$ the module $N$ when treated as an $A$-module via the map $\phi$. As usual, let $\widetilde{N}$ denote the sheaf of $\mathcal{O}_{Y}$-modules corresponding to $N$.
I have proved the following, so throughout my question it can be assumed:
Theorem 1: If $M$ is an $A$-module, with other notation from above, then there is an adjunction between restriction of scalars and change of rings. In particular, there is a natural bijection, $$ \hom_{A}(M, (_{A}N) ) \simeq \hom_{B}(B \otimes_{A} M, N). $$
I would like to prove that $f_{*}(\widetilde{N}) \simeq (_A{N})^{\sim}$. This is Hartshorne Proposition II 5.2 (d). There he claims that this "follows directly from the definitions", although I can't see how. Is anyone able to explain how it follows directly from the definition?
In any case, I have been attempting to prove this via adjunction arguments and the Yoneda lemma, and I wanted to know if people could help me finish my attempt, and also tell me if such a method is even valid. To begin, for any sheaf of $\mathcal{O}_{X}$-modules $\mathcal{F}$, define the presheaf $$ \mathcal{T}_{\mathcal{F}}: U \mapsto f^{-1} \mathcal{F}(U) \otimes_{f^{-1}\mathcal{O}_{X}(U)} \mathcal{O}_{Y}. $$ Then the sheafification of this, denoted $\mathcal{T}_{\mathcal{F}}^{+}$ is precisely the inverse image sheaf $f^{*} \mathcal{F}$.
I claim that in order to prove what I want, Yoneda's lemma tells me it is sufficient to find a natural bijection $$ \hom_{\mathcal{O}_{X}} \big( \mathcal{F}, (_AN)^{\sim} \big) \simeq \hom_{\mathcal{O}_{X}} \big( \mathcal{F} , f_{*}(\widetilde{N}) \big). $$ To do this, we begin with the well known adjunction $f^{*} \dashv f_{*}$ to write $$ \hom_{\mathcal{O}_{X}} \big( \mathcal{F} , f_{*}(\widetilde{N}) \big) \simeq \hom_{\mathcal{O}_{Y}} \big( f^{*}\mathcal{F}, \widetilde{N} \big). $$ Then by the universal property for sheafification, we have another natural bijection, $$ \hom_{\mathcal{O}_{Y}} \big( f^{*}\mathcal{F}, \widetilde{N} \big) \simeq \hom \big( \mathcal{T}_{\mathcal{F}}, \widetilde{N} \big). $$ At this point I'd like to make some argument on global sections using Theorem 1 above. Is anyone able to give me an idea how to finish this, and whether or not this is even valid up to this point? I am new to Yoneda's lemma and adjoints so I'm not entirely comfortable that it's even correct.
I mentioned at the top that a question has been asked on this result before. The reason I am asking this here is that the answer given to that question uses the fact that there is an equivalence of categories $\mathsf{Mod}A \simeq \mathsf{QCoh}(X)$. However, this result is not proved until later in Hartshorne, and indeed the proof seems to use the very fact I am trying to prove here.
