Let $X$, $Y$ be (absolute) CW-complexes and $\varphi:X\to Y$ a continuous map. I would like to know under which assumptions $(Y,\varphi(X))$ is a relative CW-complex.
I got interested in this question in a more specific case, i.e. when $Y=X$ and $\varphi$ is homotopic to the identity of $X$. However, I don't have many knowledge on algebraic topology to know related results, so every help would be appreciated.
P.S. I already know that this works if $\varphi(X)$ is a subcomplex of $Y$.