Let $S : C\rightarrow D$ and $T : D \rightarrow C$ be covariant functors and $α : S \rightarrow T$ a natural equivalence. Show that there is a natural equivalence $β : T \rightarrow S$ such that $βα = I_S$ and $βα = I_T$ , where $I_S : S \rightarrow S$ is the identity natural equivalence, and similarly for $I_T$.
I think there is a problem with that functors. I got in my script that natural equivalence works on functors which have same domain and codomain. If you explain it to me, you can give me a hint how to solve this problem. Thank you.