Pullback of a sheaf of modules ($f^{\ast }\mathcal{E}$)

For a morphism of ringed spaces, the pullback of an ${\mathcal{O}}_Y$-module is

For a morphism of ringed spaces, there is a natural morphismIt is an isomorphism when $\mathcal{E}$ is a locally free sheaf of finite rank, because the claim is local and then reduces to distributivity over a finite direct sum.