Let $X,Y$ be compact, connected, simply-connected, and separable, metric spaces each with at-least $2$-points, and let $f,g:X\rightarrow Y$ be continuous functions. Does there always exist a homeomorphism $\Phi:X\times Y \rightarrow X\times Y$ such that $$ g(x) =\pi_Y\circ \Phi(x,f(x)) $$ for all $x \in X$, where $\pi_Y$ is the canonical projection of $X$ onto $Y$?
Heuristically (but not exactly): "Can we always perturb the graph of a continuous function so that it becomes the graph of another function?"