Relative homotopy and composition of maps 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. 
 A: 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 $Y = I$.
A: This is a comment on the area of relative homotopy and composition of maps, and a positive answer to a somewhat different question, which I hope is helpful.  
Let $i: A \to X$ be a closed inclusion, and let $u: A \to Y$ be a map. Write $[(X,i),(Y,u)]$ for the set of homotopy classes rel $A$ of maps $X \to Y$ which extend $u$. Then a map $q: Y \to Z$ induces by composition a function $$q_*:[(X,i), (Y,u)] \to [(X,i),(Z,qu)]. $$
Then 7.2.6 of Topology and Groupoids (T&G) states that if $i: A \to X$ is a closed cofibration and $q: Y \to Z$ is a homotopy equivalence then the above function $q_*$ is a bijection. 
This generalises a standard  result that a homotopy equivalence $q: Y \to Z$, not necessarily pointed,  of spaces induces an isomorphism $q_*:\pi_n(Y,y) \to \pi_n(Z,q(y))$ of the homotopy groups of pointed spaces. 
As in that case, the proof involves showing that, using the cofibration condition on $i$,  a homotopy $\theta: u \simeq v$ induces a bijection $$\theta_\#: [(X,i],(Y,u)] \to [(X,i),(Y,v)].$$
(Are there other sources for this fact?) 
I think the questioner  can easily tease out why this is consistent with the examples given by k.stm. 
