Let $X$ be a scheme with structure sheaf $\mathcal{O}_X$. Then for $\mathcal{O}_X$-modules $\mathcal{F},\mathcal{M}, \mathcal{N}$ there exist a natural Hom- Tensor adjunction
$$ Hom_{\mathcal O_X}(\mathcal{F}, Hom_{\mathcal O_X} (\mathcal{M}, \mathcal{N})) = Hom_{\mathcal O_X}(\mathcal{F} \otimes_{\mathcal O_X} \mathcal{M}, \mathcal{N})$$
My question is:
If we consider the derivated functors $Ext_X ^n(\mathcal{F}, -)$ of the functor $Hom_{\mathcal O_X}(\mathcal{F}, -)$, does hold the similar formula
$$Ext_{X} ^n(\mathcal{F}, Hom_{\mathcal O_X} (\mathcal{M}, \mathcal{N})) = Ext_{X} ^n(\mathcal{F} \otimes_{\mathcal O_X} \mathcal{M}, \mathcal{N})$$
?
Background of my question is the red tagged line in following excerpt from "Vector Bundles on Complex Projective Spaces" by Christian Okonek, Michael Schneider, Heinz Spindler:
$Ext_{\mathbb{P}^1}^1(F, \mathcal{O}_{\mathbb{P}^1})=H^1(\mathbb{P}^1, F^* \otimes \mathcal{O}_{\mathbb{P}^1})$?
My ideas:
I know because of $F^*= Hom_{\mathbb{P}^1}(F,\mathcal{O}_{\mathbb{P}^1} )$ that $$H^1(\mathbb{P}^1, F^* \otimes \mathcal{O}_{\mathbb{P}^1})= Ext_{\mathbb{P}^1}^1(\mathcal{O}_{\mathbb{P}^1}, F^*)= Ext_{\mathbb{P}^1}^1(\mathcal{O}_{\mathbb{P}^1}, Hom_{\mathbb{P}^1}(F,\mathcal{O}_{\mathbb{P}^1} ))$$
Futhermore $$Ext_{\mathbb{P}^1}^1(F, \mathcal{O}_{\mathbb{P}^1})= Ext_{\mathbb{P}^1}^1(F \otimes \mathcal{O}_{\mathbb{P}^1}, \mathcal{O}_{\mathbb{P}^1})$$
To conlude $Ext_{\mathbb{P}^1}^1(F, \mathcal{O}_{\mathbb{P}^1})=H^1(\mathbb{P}^1, F^* \otimes \mathcal{O}_{\mathbb{P}^1})$? it would be nice if $Ext_{X} ^n(\mathcal{F}, Hom_{\mathcal O_X} (\mathcal{M}, \mathcal{N})) = Ext_{X} ^n(\mathcal{F} \otimes_{\mathcal O_X} \mathcal{M}, \mathcal{N})$ holds.
Does it? If yes, how to see it?