Consider two topological spaces $X$ and $Y$.
The notion of homotopy equivalence between $X$ and $Y$ is defined as a pair of continuous maps $f:X\to Y$ and $g:Y\to X$ such that $f\circ g$ and $g\circ f$ are homotopic to the identities.
But what if we weaken the definition a little bit, by considering the following "weaker" homotopy equivalence between $X$ and $Y$ : it is defined as a pair of pairs of continuous maps, $(f,g)$ and $(f',g')$, where $f,f':X\to Y$, and $g,g':Y\to X$ such that $g\circ f$ is homotopic to $Id_X$ and $f'\circ g'$ is homotopic to $Id_Y$.
I'm asking this question because I don't see why intuitively we would require $f'$ and $g'$ to be actually equal to $f$ and $g$.
Would this definition give an interesting notion of homotopy equivalence, or would this just coincide with the usual definition ? Or would this be simply too weak to do anything with it ?