# Show that if $I$ and $I'$ are initial objects there exists a unique isomorphism between them.

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 $$I'$$.

You're halfway through, because you didn't show these arrows going in opposite directions are actually inverses. Assume $$I$$ and $$I'$$ are initial. Then since $$I$$ is initial, there is a unique arrow $$f:I \to I'$$. Since $$I'$$ is initial, there is a unique arrow $$g:I' \to I$$. Now, both $$g \circ f$$ and $$1_I$$ are arrows $$I \to I$$, but since $$I$$ is initial, there can be only one: we conclude that $$g\circ f = 1_I$$. Similarly, $$f \circ g = 1_{I'}$$. Now you can say that $$I \cong I'$$.