I am currently reading a paper where the author needs to show that a map $i: X \hookrightarrow Y$ between 2 $CW$- complexes $X$,$Y$ (and $X$ is contained in $Y$) is a deformation retraction.
He claims it suffice to show that for every $\psi : D^n \hookrightarrow Y$ such that $\psi(\partial D) \subset X$, we can produce a deformation retraction of $\psi(D)$ to a disk contained in $X$ while fixing $\psi(\partial D)$.
Why is this equivalent to showing that there is a deformation retraction from $Y$ to $X$ ?