Suppose $X\subseteq P^n$ be a quasiprojective variety with $f:X\rightarrow\mathbb P^m$ a regular map. Can anyone please help me to show that $(f,id):X\times\mathbb P^m\rightarrow\mathbb P^m\times\mathbb P^m $ is a regular map. I am following Shaferevich's definition of regular maps between quasiprojective varieties.
In general is it true that if $X_1,X_2, Y_1, Y_2$ are quasiprojective varieties and $f_1:X_1\rightarrow Y_1$ and $f_2:X_2\rightarrow Y_2$ are regular maps then their product is also a regular map?
$\textbf{Edit:}$ Let $\phi_1: X\times\mathbb P^m\rightarrow\mathbb P^{N_1}$ be the segre embedding and $\phi_2: \mathbb P^m\times\mathbb P^m\rightarrow\mathbb P^{N_2}$ be segre embedding. I need to show that $\phi_2\circ(f,id)\circ \phi_1^{-1}: \phi_1(X\times\mathbb P^m)\rightarrow\phi_2(\mathbb P^m\times\mathbb P^m)$ is a regular map. How to show this?