I recently learned about the concept of a Hurewicz fibration. I tried to prove that for a product of topological spaces $X \times Y$, the projection $p: X \times Y \rightarrow X$ is a Hurewicz fibration, something that on wikipedia is described as 'very easily seen'. I thought that maybe I should somehow use the universal propery of the product, but I still don't see how to proceed after setting up the commutative diagram
$$ \require{AMScd} \begin{CD} Z @>g>> X \times Y\\ @V{i_0}VV @VVpV \\ Z \times I @>>h> X \end{CD} $$
in order to obtain a lifting morphism $d: Z \times I \rightarrow X \times Y$. Could someone give me some suggestions, or a reference to a resource where this fact is proven?
Let $g: Z \to X \times Y$ be a given map, with a specified homotopy $h: Z \times I \to X$ such that $h(z,0) = p(g(z))$ for all $z$.
Then, we can lift the homotopy to $d: Z \times I \to X \times Y$ by $d(z,t) := (h(z,t), \pi_2(g(z)))$.