Tensor compatibility of graded sheafification on degree-one-generated Proj ($\tilde{M}{\otimes }_{{\mathcal{O}}_X}\tilde{N}\cong \widetilde{M{\otimes }_{S}N}$)

Under the degree-one generation condition for Proj, cover $X=\mathrm{Proj}S$ by degree-one charts. The degree-one localization of a graded module identifies the two local modules in the displayed formula, using localization commutes with tensor products. The maps are natural and agree on overlaps, so they glue to an isomorphism of sheaves of modules. The analogous assertion can fail for general positively graded rings.