I'm having some trouble wrapping up the proof for 16.3.A in Vakil's FOAG, which basically asks to show that the tensor product construction of $\pi^{\ast} \mathscr{G}$ (for $\pi : X \to Y$, $\mathscr{G}$ quasicoherent on $Y$) satisfies the following universal property: for any $\mathscr{O}_{X}$ module $\mathscr{F}$ (not necessearily quasicoherent) there is a (functorial) bijection:
$$ \operatorname{Hom}_{\mathscr{O}_X}(\pi^{\ast} \mathscr{G}, \mathscr{F}) $$
$$ \updownarrow $$
$$ \operatorname{Hom}_{ \mathscr{O}_Y }( \mathscr{G} , \pi {\ast} \mathscr{F} ) $$
In fact question 16.3.A simplifies this by assuming that $Y = \operatorname{Spec} B$ is affine. If I fix an affine in the preimage, say $\pi (\operatorname{Spec} A) \subset \operatorname{Spec} B$, general intuition from the prior sections (and the questions hint) tell me to use the adjointness of $(\cdot \otimes_B A, \cdot_B )$ on the level of modules / rings. By Exercise 2.5.C in Vakil's text (or Theorem 13.3.2, whichever you prefer), we may work on the level of a distinguished affine base in order to work our way down to this case of adjointness.
Let $ \mathscr{G} = \widetilde{N} $ so that $ \pi^\ast \mathscr{G} \vert_{\operatorname{Spec} A} = \widetilde{ N \otimes_B A} $ and fix some $\mathscr{O}_X$-linear $\phi : \pi^\ast \mathscr{G} \to \mathscr{F}$; we should have on each distinguished open affine $D(f) \subset \operatorname{Spec} A$ the data of an $A_f$-linear map $$ \phi(D(f)) : N \otimes_B A_f \to \mathscr{F}(D(f)) $$
My thought was to use adjointness of $\cdot \otimes_B A_f$ here, except in this case the functor $\cdot_B$ arises from the composition of ring maps $B \to A \to A_f$. Then we would get a unique map $ \psi : N \to \mathscr{F}(D(f))_B $; however, the issue is that this doesnt really carry local data well: if we have our corresponding morphism of local rings sends some $g \in B$ to $f \in A$, I can imagine we would somehow want adjointness to give a map $$ \psi(D(g)) : N_g \to \mathscr{F}( \pi^{-1}(D(g)) ) $$
which would then glue (using Exercise 2.5.C / Theorem 13.3.2 again) to get a morphism of sheaves.
In fact, it's not entirely clear to me how to play with $\mathscr{F}$ here since it is not quasicoherent; exercise 16.2.A only tells us how to associate $\pi_\ast \widetilde{M}$ and $\widetilde{M}_B$, but nothing from §16.2 tells us what to do in the case of more general $\mathscr{O}_X$-modules.
Any help would be appreciated. (Also for any admin I cant seem to get that functorial bijection above to render correctly into one MathJax cell — the last term is meant to read $\pi_\ast \mathscr{F}$ but I suspect certain underscores are being rendered as markdown).
