How does this product of matrices define a local diffeomorphism?

Question

Let $$H_1 := \left\{\begin{pmatrix} 1 & x \\ 0 & 1\end{pmatrix} \mid x \in \mathbb{R}\right\},$$ $$H_2 := \left\{\begin{pmatrix} 1 & 0 \\ y & 1\end{pmatrix} \mid y \in \mathbb{R}\right\},$$ and $$H_3 := \left\{\begin{pmatrix} \lambda & 0 \\ 0 & \lambda\end{pmatrix} \mid \lambda \in \mathbb{R}^\ast\right\}.$$ How does the product map $H_2 \times H_3 \times H_1 \rightarrow \operatorname{SL}(2, \mathbb{R})$ define a local diffeomorphism?