Sheaf Hom between locally free sheaves

Let $\{U_i\}_{i \in I}, \{V_i\}_{j \in J}$ be covers of $X$ such that $$\mathcal F|_{U_i} \cong \mathcal O_{U_i}^{\oplus m}, \quad \mathcal G|_{V_j} \cong \mathcal O_{V_j}^{\oplus n}.$$

Then $\{U_i \cap V_j\}_{i\in I, j \in J}$ is a cover of $X$ such that $$\mathcal F|_{U_i \cap V_j} \cong \mathcal O_{U_i \cap V_j}^{\oplus m}, \quad \mathcal G|_{U_i \cap V_j} \cong \mathcal O_{U_i \cap V_j}^{\oplus n}.$$

For any open $W \subset U_i \cap V_j$, $$\mathcal{Hom}(\mathcal F, \mathcal G)(W)=\operatorname{Hom}(\mathcal F|_{W}, \mathcal G|_{W})\cong \operatorname{Hom}(\mathcal O_{W}^{\oplus m},\mathcal O_{W}^{\oplus n}) \cong \operatorname{Hom}(\mathcal O_{W},\mathcal O_{W})^{\oplus mn} \cong \mathcal O_W^{\oplus mn}.$$

So, $$\mathcal{Hom}(\mathcal F, \mathcal G)|_{U_i \cap V_j} \cong \mathcal O_{U_i \cap V_j}^{\oplus mn}.$$