I'm facing the following problem and have no idea how to deal with it.
We consider a functor $T:\underline{Set}\rightarrow\underline{Set}$ and two sets $X,Y$. We can build the product $X\times Y$ and calculate $T(X\times Y)$. On the other hand we can built $T(X)\times T(Y)$.
If for example $T$ is the powerset functor, then $T(X\times Y)$ is larger than $T(X)\times T(Y)$ and there is a surjection $t:T(X\times Y)\twoheadrightarrow T(X)\times T(Y)$.
If $T$ is an arbitrary set functor, is there some sort of general abstract criterion or a categorical property of $T$, assuring the existence of such a surjection $t$?