Remark 4.2.4.4. Let $f: A \rightarrow B$ be a morphism of simplicial sets. If $f$ is either left or right anodyne, then it is anodyne (Definition 3.1.2.1). In particular, any left or right anodyne morphism of simplicial sets is a monomorphism (Remark 3.1.2.3) and a weak homotopy equivalence (Proposition 3.1.6.14). Conversely, if $f$ is inner anodyne (Definition 1.5.6.4), then it is both left anodyne and right anodyne. That is, we have inclusions
\[ \xymatrix@C =30pt@R=30pt{ \{ \textnormal{Inner anodyne morphisms} \} \ar@ {}[d]|{\cap } \ar@ {}[r]|{\subset } & \{ \textnormal{Left anodyne morphisms} \} \ar@ {}[d]|{\cap } \\ \{ \textnormal{Right anodyne morphisms} \} \ar@ {}[r]|{\subset } & \{ \textnormal{Anodyne morphisms} \} . } \]
All of these inclusions are strict (see Example 4.2.4.7).