Do fog and gof uniquely determine f and g?

Question

If fog and gof are given/known, then are f and g uniquely determined by these? If yes, then how to find them? I am clueless about the general case.

Answer 1

Of course NO! Consider any bijection $f:\mathbb{R}\rightarrow \mathbb{R}$. Put $g = f^{-1}$. Then $f\circ g$ and $g\circ f$ are identity maps on $\mathbb{R}$. Clearly we cannot recover $f$ and $g$ by just knowing $f\circ g$ and $g\circ f$.