Let $X$ be a set and $T$ be a compact metric space. $f:X\times X\to T$ is a function.
There are two ways for $f$ to be considered as a function from $X$ to the function space $T^X$ (it is a metric space with $d(f,g)=\sup_i (d(f(i),g(i)))$ ) , that is $f_1(x)= (y \mapsto f(x,y))$ and $f_2(y)= (x \mapsto f(x,y))$.
Prove that the image of $f_1$ is precompact (closure is compact) iff the image of $f_2$ is precompact.