I am trying to prove something and am stuck on the following issue :
Suppose $\Psi, \Phi : I^n \to Y$ are two maps and $q:Y \to Z$ is a homotopy equivalence such that $q \Phi \cong q \Psi $rel $\partial I^n $. Can we conclude that $\Phi \cong \Psi $rel $\partial I^n $ ?
I know that if we were talking about ordinary homotopy (without relative to $\partial I^n$) then the answer is affirmative. But am unable to proceed in this case.
A slightly modified question is :
Suppose $\Lambda, \Xi: I^n \to Y $are two maps and $ \mu : Y \to Y $ is homotopic to identity. Suppose further that $\mu \Lambda \cong \mu \Xi $ rel $ \partial I^n$. Can we conclude that $ \Lambda \cong \Xi $ rel $ \partial I^n$ ?
An affirmative answer to the second question implies an affirmative answer to the first. I appreciate any help.
For the first one, let $Y = I$ and $Z = \star$, the singleton space, so that $q \colon Y → Z$ is the constant map.
Let furthermore $Φ$ and $Ψ$ be different constant maps $I^n → Y$. Then $Φ$ and $Ψ$ cannot be homotopic relative to any non-empty subspace, but $qΦ = qΨ$.
For the second one, take the first counterexample while regarding $Z = \star$ as a subspace of $Y = I$.