I am wondering if there is a relation between a map $(\varphi, \psi) \in \rm{Diff}(\mathbb{R}^n) \times \rm{Diff}(\mathbb{R}^m)$ and a map $\Psi \in \rm{Diff}(\mathbb{R}^n \times \mathbb{R}^m)$. To be clear: Here I only mean local diffeomorphisms at $(0,0)$. Is there an embedding of $\rm{Diff}(\mathbb{R}^n) \times \rm{Diff}(\mathbb{R}^m)$ into $\rm{Diff}(\mathbb{R}^n \times \mathbb{R}^m)$, since maps $(\varphi, \psi)\in \rm{Diff}(\mathbb{R}^n) \times \rm{Diff}(\mathbb{R}^m)$ are of the form $(x,y) \mapsto (\varphi(x),\psi(y))$ and a map $\Psi \in \rm{Diff}(\mathbb{R}^n \times \mathbb{R}^m)$ is of the form $\Psi(x,y) = (\Psi_1(x,y), \Psi_2(x,y))$. Can I understand the map $\Psi_1$ as a parametrized diffeomorphism, i.e. for all $y \in \mathbb R^m$ the map $x \mapsto \Psi_1(x,y)$ is a diffeomorphism in $\mathbb R^n$ and vice versa for $\Psi_2$?
Thanks in advance, any help is appreciated.
EDIT: So I figured that $\rm{Diff}(\mathbb{R}^n) \times \rm{Diff}(\mathbb{R}^m)$ is embedded in $\rm{Diff}(\mathbb{R}^n \times \mathbb{R}^m)$. As for $(\varphi, \psi) \in \rm{Diff}(\mathbb{R}^n) \times \rm{Diff}(\mathbb{R}^m)$, the map $\iota:(\varphi, \psi) \mapsto (\varphi \times \psi)$ is clearly injective and continuous, therefore an embedding and $(\varphi \times \psi)\in \rm{Diff}(\mathbb{R}^n \times \mathbb{R}^m)$.
Is this correct or am I missing something?