If $(X,A)$ has homotopy extension property, then $(X \times I, X \times \partial I \cup A \times I)$ also shares this property.

Question

The original problem is to prove:

If $(X,A)$ has HEP (homotopy extension property), then $(X \times I, X \times \partial I \cup A \times I)$ also shares this property.

I found a proof in Page 35, Theorem 2.33 of this note, or see here in picture, but I want an explicit one.

My question:

Can we get an explicit expression of retraction $$\phi: X \times I \times I \to X \times I \times \{0\} \cup (X \times \partial I \cup A \times I) \times I$$ ?

Thank you!