Suppose $X,Y$ are topological spaces with $Y$ homogeneous and $f,g:X\to Y$ continuous such that there exist continuous functions $u,v:Y\to Y$ such that $$f = u\circ g \text{ and } g= v \circ f.$$

Does this imply that there is a homeomorphism $\varphi:Y\to Y$ such that $f = \varphi \circ g$?

**Note.** When the homogeneity condition on $Y$ is dropped, there are counterexamples: Let $y,z\in Y$ such that there is no homeomorphism on $Y$ sending $y$ to $z$, and let $f$ be the constant function from $X$ sending everything to $y$, and let $g$ be the constant function from $X$ sending everything to $z$.