Skeletal categories and isomorphisms between an object and itself Is the identity arrow the unique isomorphism between an object and itself in a skeletal category? I tried to show that two skeletons $C_0$ and $C_1$ in a given category $C$ are isomorphic. I chose F the functor that maps an element of $C_0$, viewed as an element of $C$, to its unique correspondant in $C_1$, and similarly for $G$ with an element of $C_1$, viewed as an element of $C$, onto its unique correspondant in $C_0$. I want to show that $F$ composed with $G$ is the unit functor in $C_1$, but for a given arrow g in $C_1$, I show that $F$ composed with $G$ has same domain and codomain as $g$ but can't seem to show immediately that it is $g$ itself.

The comments already answered the question negatively, so I'm giving a simple counterexample for when the skeleton possesses non-trivial automorphisms.
For $\mathsf{FinVect}_\mathbb R$ the category of finite dimensional real vector spaces, choosing basss for each $V$ we get isomorphisms $V\cong \mathbb R^{\dim V}$.
This means that $\mathsf{sk(FinVect}_\mathbb R)$ can be described by the $\mathbb R^n$ and linear transformations between them. And of course there are other linear transformations $\mathbb R^n\to\mathbb R^n$ other than the identity.
In fact, if any $x\in\mathsf C$ has a non-trivial automorphism then this is inherited in $\mathsf{sk(C)}$.
