Direct-image tensor comparison ($f_{\ast }\mathcal{F}{\otimes }_{{\mathcal{O}}_Y}{f}_{\ast }\mathcal{G}\to f_{\ast }(\mathcal{F}{\otimes }_{{\mathcal{O}}_X}\mathcal{G})$)

Tensoring sections on $f^{-1}V$ gives a balanced pairing of the two direct image sheaves on $V$. The universal property of the tensor product of modules and sheafification produce the displayed morphism. Composing with the pullback-direct-image adjunction unit gives the projection formula for sheaves. The comparison itself need not be an isomorphism.