In an exercise of Hartshorne, we are to prove that ($f : X \to Y$ a morphism of schemes) pullback $f^*$ induces a morphism $\text{Pic }Y \to \text{Pic }X$ on the group of invertible sheaves under the operation of tensor product. In order to show that $f^*(L_1 \otimes L_2) \cong f^*L_1 \otimes f^*L_2$, I should think there would be an easy application of Yoneda's lemma, if only I could find the bijection
$$\text{Hom}_{\mathcal{O}_X}(f^*(L_1 \otimes L_2), \mathcal{F}) = \text{Hom}_{\mathcal{O}_X}(f^*L_1 \otimes f^*L_2, \mathcal{F})$$
Starting with adjointness on the left, I get
$$\text{Hom}_{\mathcal{O}_X}(f^*(L_1 \otimes L_2), \mathcal{F}) = \text{Hom}_{\mathcal{O}_Y}(L_1 \otimes L_2, f_*\mathcal{F}) = \text{Hom}_{\mathcal{O}_Y}(L_1, \text{Hom}_{\mathcal{O}_X}(L_2, f_*\mathcal{F}))$$
but I'm not sure where to go from there.
(On a possibly related note, I realize the notation $\text{Hom}_{\mathcal{O}_X}(\mathcal{M}, f_*\mathcal{N})$ as a set really only refers to the global sections of $\text{Hom}_{\mathcal{O}_X}(\mathcal{M}, f_*\mathcal{N})$ as an $\mathcal{O}_X$-module, but seeing as the former is in bijection with the set $\text{Hom}_{\mathcal{O}_Y}(f^*\mathcal{M}, \mathcal{N})$, is there a categorical relationship between the two realized as modules of sheaves?)
You can go on with your chain of isomorphisms using that $$\mathcal{Hom}_{\mathcal{O}_Y}(\mathcal{G},f_{\ast}\mathcal{F})\cong f_{\ast}\mathcal{Hom}_{\mathcal{O}_X}(f^{\ast}\mathcal{G},\mathcal{F}).$$ To contstruct this isomorphism, just look at the sections of the sheaves and use the adjunction.