Yes, this follows e.g. from the socalled projection formula: if $f:X\to Y$ is a morphism of ringed spaces, $\mathcal F$ an $\mathcal O_X$-module, and $E$ be a locally-free $\mathcal O_Y$ module of finite rank, then $f_* (\mathcal F \otimes f^{*} E) \simeq f_{*}\mathcal F \otimes E$.