Commutativity of Diagram in Top "Lift-like property but for covers"

Question

Let $\psi,\chi,\phi$ be continuous functions, with $\psi,\xi$ surjective. Does there necessarily exists a continuous function $\eta$ such that the diagram commutes in Top: $$\require{AMScd} \begin{CD} A @>{\phi}>> B\\ @V{\psi}VV @V{\xi}VV \\ C @>{\eta}>> D \end{CD}$$

Answer 1

This is not true even just for functions. Indeed, if $a,b\in A$ and $c\in C$ are such that $\psi(a)=\psi(b)=c$ but $\xi(\phi(a))\neq \xi(\phi(b))$, then there is no choice of $\eta(c)$ that will make the diagram commute.