I'm revising for my exams now and struggling with the following exercise:
Let $f: X \to S$ be an open or closed immersion and $g: S' \to S$ another morphism where $X,S,S'$ are schemes. Then the base-change morphism $f' : X \times_S S' \to S'$ is such that $\text{Im}(f') = g^{-1}(\text{Im}(f))$.
If anyone could prove this for me I'd be really grateful!