Homotopy equivalence between pointed CW-complexes

Question

Let $f : (X, x_0) \to (Y, y_0)$ a morphism in the category of pointed, connected CW-complexes. Let $A= [1/2, 1]^{+} \wedge X \cup Y \subset C_f$, where $C_f$ is the mapping cone and $X, Y$ are pointed, connected CW-complexes.

I was trying to prove that $A$ is homotopy equivalent to $Y$: I am pretty sure it is the inclusion of $Y$ in $A$ that determines the equivalence, but I have trouble finding an explicit expression of it.

Any hints?

Answer 1

$A$ is a copy of the reduced mapping cylinder of $f$ of which $Y$ is a strong deformation retract.