$\operatorname{Hom}_{\mathcal{A}}(\mathcal{M,N}) \simeq \operatorname{Hom}_{\mathcal{B}}(\mathcal{M} \otimes_{\mathcal{A}} \mathcal{B,N})$?

Question

Let $\varphi \colon \mathcal{A} \to \mathcal{B}$ be a morphism of sheaves of ring. Let $\mathcal{M}$ be a $\mathcal{A}$-module and $\mathcal{N}$ be a $\mathcal{B}$-module. Do we have $$ \operatorname{Hom}_{\mathcal{A}}(\mathcal{M}, \mathcal{N}) \simeq \operatorname{Hom}_{\mathcal{B}}(\mathcal{M} \otimes_{\mathcal{A}} \mathcal{B}, \mathcal{N})? $$