This may be pretty trivial, but I can't figure it out. Suppose that $S$ is any scheme, and $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ are two morphisms of $S$-schemes, such that the closed image of each one exists. That is, there is a smallest closed subscheme $Z_i$ of $Y_i$ over which $f_i$ factorizes (for $i=1,2$).

Is it true then that the closed image of $f_1\times_S f_2:X_1\times_S X_2\to Y_1\times_S Y_2$ exists? In this case, is it equal to $Z_1\times_S Z_2$ (with its natural closed immersion into $Y_1\times_S Y_2$)?