First, let $\mathcal{G}$ be a groupoid. Then an automorphism $\gamma\colon X\rightarrow X$ in $\mathcal{G}$ considered as a loop in the nerve of $\mathcal{G}$ is homotopic to the point $X$ if and only if $\gamma=\mathrm{id}_X$.

This is no longer true in an arbitrary category $\mathcal{C}$. Consider for example the category consisting of two objects $X,Y$, an automorphism $\gamma\colon X\rightarrow X$ with $\gamma^{-1}=\gamma$ and another arrow $\alpha\colon X\rightarrow Y$ such that $\alpha\circ\gamma=\alpha$, i.e. $\alpha$ (co)absorbs $\gamma$. Surely, we then have the homotopy $$\xrightarrow{\gamma}\ \sim\ \xrightarrow{\gamma}\xrightarrow{\alpha}\xleftarrow{\alpha}\ \sim\ \xrightarrow{\alpha\circ\gamma}\xleftarrow{\alpha}\ \sim\ \xrightarrow{\alpha}\xleftarrow{\alpha}\ \sim\ X$$ Similarly, if we reverse the direction of the arrow $\alpha$, we have an arrow which absorbs $\gamma$.

**Question:** Can you provide an example of a category with a non-trivial automorphism which is neither absorbed nor co-absorbed by another arrow but is null-homotopic.