Homotopy and “transfer” equivalence


Let $ X,Y$ be topological spaces and $ f,g:X\to Y$ be continuous. We say that there is a transfer function from $ f$ to $ g$ if there is $ u:Y\to Y$ such that $ g = u\circ f$ , and we write $ g\leq_t f$ . We say that $ f,g$ are transfer equivalent if $ g\leq_t f$ and $ f\leq_t g$ .

What is an example of spaces $ X,Y$ and continuous functions $ f,g:X\to Y$ such that $ f, g$ are homotopic, but not transfer equivalent – and what is an example for continuous functions that are transfer equivalent, but not homotopic?