Existence of two element Partitions for Sets X and Y where Functions f and g map partition elements to each others

Question

Let $X$ and Y be two non-empty sets and f be a function from X to Y and g be a function from Y to X. We need to prove that there are partitions (A,B) of X, and (C,D) of Y such that f(A)=C and g(D)=B. As the functions f and g are arbitrary and there is no connection between them, Is this always possible?