 Solution

An element of the left side is $f(s)$ for some $s\in S$ with $f(s)\in T$, exactly an element of $T\cap f(S)$. The second equality need not hold, even for a surjection: map $\{1,2\}$ to a singleton and take $S=\{1\}$, $S^{\prime }=\{2\}$. Then the image of the intersection is empty but the intersection of the images is not.