This question is borne out of the following claim: Let $X = \text{ Spec }A$ be an affine scheme with $M$ and $N$ $A$-modules. Then $(M \otimes_{A} N)^{\sim} \simeq \widetilde{M} \otimes_{\mathcal{O}_{X}} \widetilde{N}$, as well as the similar claim that for a family $\{ M_{i} \}_{i \in I}$ of $A$-modules, then we have $\oplus_{i} \widetilde{M}_{i} \simeq \oplus_{i} (M_{i} )^{\sim}$.
Hartshorne claims this is obvious since tensor products and direct sums commute with localization. However, that this implies the result is not at all obvious to me. Certainly this gives us agreement on stalks, and even agreement on distinguished affine open sets. However, unless this agreement is induced by a morphism of sheaves, then this does not say that the sheaves are isomorphic. Take the tensor product case. Define the presheaf $$ \mathcal{T}: U \mapsto \widetilde{M}(U) \otimes_{\mathcal{O}_{X}(U)} \widetilde{N}(U). $$ Then the sheafification of this is the tensor product sheaf. So by the universal property for sheafification, I need only construct an isomorphism $$ \mathcal{T} \longrightarrow (M \otimes_{A} N)^{\sim}. $$ On distinguished affine open sets, it is easy to see that these two (pre)sheaves agree, indeed: $$ \mathcal{T}(D(h)) \simeq M_{h} \otimes_{A_{h}} N_{h} \simeq (M \otimes_{A} N)^{\sim}(D(h)), $$ but how do I construct a morphism on which these isomorphisms are precisely the morphisms of sections?
I am also wondering about what is more or less a converse problem. Someone claimed that the above follows easily from the adjunction of functors $$ \text{Hom}_{A}(M, \Gamma(X, \mathcal{F} )) \simeq \text{Hom}_{A}(\widetilde{M},\mathcal{F}) $$ between $M \mapsto \widetilde{M}$ and $\mathcal{F} \mapsto \Gamma(X, \mathcal{F})$. However, while this certainly gives me the existence of a morphism between the sheaves I want, I have no explicit description of the morphism, and hence no way to check that this is an isomorphism on stalks.
Is someone able to tell me what the "best" way to show that isomorphism for the tensor product and direct sum is? In the first case, how do I show that the isomorphism on stalks actually arises from a morphism of sheaves, and in the second case, how do I show that the morphism of sheaves gives an isomorphism on stalks?