Let $X$ and $Y$ be topological spaces. Suppose we have an isotopy between maps $f, g: X\to Y$.
The question is that is there a homeomorphism $h: Y\to Y$ such that $h\circ f =g$?
I am especially interested in the case when $Y$ is a surface and $f, g$ are embeddings.
Is this true or do we need more conditions? Is there any analogous result?
Thank you in advance.