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?
Let $ \varphi:X\times X'\rightarrow \mathbb{P}^N $ and $ \psi:Y\times Y'\rightarrow \mathbb{P}^M $ be the Segre embeddings. To prove that $ f\times f' $ is a regular map, it means that to prove the map $ \psi\circ (f\times f')\circ \varphi^{-1}:\varphi(X\times Y)\rightarrow \psi(X'\times Y') $ is a regular map. For all point $ (x,y)\in X\times Y $, there exist neighborhoods contained in affine spaces $ V $ of $ f(x) $ and $ V' $ of $ f'(y) $ s.t. $ \psi:V\times V'\rightarrow \psi(V\times V') $ is an isomorphism. Since $ f,f' $ are continuous, there exist neighborhoods contained in affine spaces $ U $ of $ x $ and $ U' $ of $ y $ s.t. $ f(U)\subset V $ and $ f'(U')\subset V' $ and $ \varphi:U\times U'\rightarrow \varphi(U\times U') $ is an isomorphism. Hence it suffices to prove that $ (f,f'):U\times U'\rightarrow V\times V' $ is a regular map, i.e., we can assume that $ X,X' $ and $ Y,Y' $ are quasiprojectivies contained in affine spaces. But in that case, it is clear that $ (f,f') $ is a regular map. This completes the proof.