Let $(X,\mathcal O_X)$ be a separated Noetherian scheme. Let $\mathcal E, \mathcal F, \mathcal G$ be coherent sheaves such that $\mathcal E, \mathcal F$ are algebraic vector-bundles (i.e. locally free coherent sheaves). Then, do the following isomorphisms hold:
(1) $$\mathcal Hom_{\mathcal O_X}(\mathcal E, \mathcal G)\otimes_{\mathcal O_X}\mathcal F \cong \mathcal Hom_{\mathcal O_X}(\mathcal E, \mathcal G\otimes_{\mathcal O_X}\mathcal F)$$
(2) $$\mathcal E \otimes_{\mathcal O_X}\mathcal Hom_{\mathcal O_X}(\mathcal F,\mathcal G)\cong \mathcal Hom_{\mathcal O_X}(\mathcal Hom_{\mathcal O_X}(\mathcal E, \mathcal F),\mathcal G) $$ ?
My motivation: I can prove both of these if $X$ is affine-scheme (in fact there I only need $\mathcal E$ to be vector-bundle i.e. I need it's global section module to be projective)