Kerodon

In the second diagram of Example 1.1.2.8, the label $\mathrm{id}_x$ and $e$ might need to be switched.

Yep. Thanks!

Remarks 1.1.2.5 and 1.1.2.6 could be expanded respectively to:

1. $f$ is monic if and only if it preserves nondegenerate simplices and for every nondegenerate simplices $x,x'$ in $X$, we have that $f(x)=f(x')$ implies $x=x'$, and
2. $f$ is epic if and only if every nondegenerate simplex in $Y$ has an inverse image along $f$,

so that combined they prove:

$f$ is an isomorphism if and only if it induces a bijection on non-degenerate simplices.

Although maybe the converse in Point 1 might fit better in Subsection 1.1.3, for its proof requires Proposition 1.1.3.8 from there.