If $\mathcal{C}$ is a category and $X,Y\in\mathrm{obj}(\mathcal{C})$, I like the notation $\mathcal{C}(X,Y)$ to denote $\mathrm{Hom}_{\mathcal{C}}(X,Y)$.
So, $\mathcal{C}(X,X)=\mathrm{End}_{\mathcal{C}}(X)$.
What do you think of the notation $\mathcal{C}(X):=\mathrm{Aut}_{\mathcal{C}}(X)$ ?
This would be consistent with the notation (or similar notations) $\mathsf{DIFF}(S^1)$ (resp. $\mathsf{TOP}(S^1)$ ) for diffeomorphisms (resp. homeomorphisms) of the circle, i.e. the $\mathrm{Aut}$ in the category $\mathsf{DIFF}$ of smooth manifolds (resp. $\mathsf{TOP}$ of topological manifolds), sometimes usedused in topology (see e.g. Andhere and here. And (see e.g. here) $\mathsf{TOP}(n)=\mathrm{Aut}_{\mathsf{TOP}}(\mathbb{R}^n)$.
