I am new to set theory and I have come across this exercise. I think the proof I have come up with is too simple to be right. would greatly appreciate some tips.
If $I$ is an initial set, then for $\forall B$ in a category C, $\exists !$ an arrow $I \to B$. Since $I'$ is a set in $C$, $\exists !$ arrow to A such that $I \to I'$. The same theory can be applied to prove that there is an arrow $B$ such that $I' \to I$. Therefore, there is an isomorphism between $I$ and $I'$.