Inverse image sheaf ($f^{-1}\mathcal{E}$)

The inverse image sheaf $f^{-1}\mathcal{E}$ is the sheaf associated with the presheaf whose sections near $U\subseteq X$ are obtained as a colimit of $\mathcal{E}(V)$ over open sets $V\supseteq f(U)$.

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.