Example 4.5.3.16 ([MR4042827]). Let $X = \Delta ^{2} \coprod _{ \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) } \Delta ^0$ be the simplicial set obtained from the standard $2$-simplex by collapsing the final edge to a point, which we represent by the diagram
Then $X$ has exactly two nondegenerate edges $e,e': \Delta ^1 \rightarrow X$, as indicated in the diagram. We now make the following observations:
There is a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{1} \ar [r] \ar [d] & \Delta ^2 \ar [d] \\ \Delta ^{1} \ar [r]^-{e'} & X. } \]Consequently, the morphism $e': \Delta ^1 \rightarrow X$ is inner anodyne, and therefore a categorical equivalence (Corollary 4.5.3.14).
There is a unique morphism of simplicial sets $r: X \rightarrow \Delta ^1$ satisfying $r \circ e' = \operatorname{id}_{\Delta ^1}$; the composite map $\Delta ^2 \twoheadrightarrow X \xrightarrow {r} \Delta ^1$ is given on vertices by $0 \mapsto 0$, $1 \mapsto 1$, and $2 \mapsto 1$. Since $e'$ is a categorical equivalence, it follows that $r$ is also a categorical equivalence (Remark 4.5.3.5).
The composite map $\Delta ^1 \xrightarrow {e} X \xrightarrow {r} \Delta ^1$ is equal to the identity map $\operatorname{id}_{\Delta ^1}$. Since $r$ is a categorical equivalence, it follows that $e$ is also a categorical equivalence. Moreover, $e$ is also a monomorphism of simplicial sets which is bijective on vertices.
The morphism $e: \Delta ^1 \hookrightarrow X$ is an inner fibration. This follows from Remark 4.1.1.5, since we have a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{2}_{2} \ar [r] \ar [d] & \Delta ^1 \ar [d]^{e} \\ \Delta ^2 \ar [r] & X, } \]where the horizontal maps are surjective and the inclusion $\Lambda ^2_{2} \hookrightarrow \Delta ^2$ is an inner fibration (since can be realized as the nerve of a morphism between partially ordered sets).
The morphism $e$ is not inner anodyne, since the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{1} \ar [d]^{e} \ar [r]^-{\operatorname{id}} & \Delta ^1 \ar [d]^{e} \\ X \ar [r]^-{\operatorname{id}} \ar@ {-->}[ur] & X } \]has no solution.