Hatcher Algebraic Topology 0.28

Exercise 0.28 in Hatcher's Algebraic Topology states

Show that if $$(X_1,A)$$ satisfies the homotopy extension property, then so does every pair $$(X_0 \sqcup_f X_1, X_0)$$ obtained by attaching $$X_1$$ to a space $$X_0$$ via a map $$f: A \to X_0$$.

This question strikes me as trivial, so I wanted to check whether my solution doesn't gloss over any important subtleties.

The homotopy extension property guarantees that there is a retract from $$X_1 \times I$$ to $$X_1 \times \{0\} \cup A \times I$$. This induces a retract from $$(X_0 \sqcup_f X_1)\times I$$ to $$(X_0 \sqcup_f X_1)\times\{0\} \cup (X_0 \sqcup_f A) \times I$$, since $$A \times I \subset X_1 \times I$$ is unaffected by the retract and hence the space $$X_0 \sqcup_f A$$ just 'spectates'. This retract implies the pair $$(X_0 \sqcup_f X_1, X_0)$$ satisfies the homotopy extension property.

Is this proof satisfactory?

A satisfactory proof would be to start with what is given, namely the retract $$r : X_1 \times I \to X_1 \times \{0\} \cup A \times I$$ next use $$r$$ to write down an actual formula, expressed in terms of the given information, for a retract $$r' : (X_0 \sqcup_f X_1) \times I \to (X_0 \sqcup_f X_1) \times \{0\} \cup (X_0 \sqcup_f A) \times I$$ and finally cite appropriate theorems as needed to guarantee continuity of $$r'$$.
The one possible source of subtlety is in that final citation: since there are quotient topologies floating around in the domain and range of $$r'$$, presumably you'll be citing the universal properties of quotient maps in justifying continuity of $$r'$$.