Remark 18.104.22.168. Let $f: A \rightarrow B$ be a morphism of simplicial sets. If $f$ is either left or right anodyne, then it is anodyne (Definition 22.214.171.124). In particular, any left or right anodyne morphism of simplicial sets is a monomorphism (Remark 126.96.36.199) and a weak homotopy equivalence (Proposition 188.8.131.52). Conversely, if $f$ is inner anodyne (Definition 184.108.40.206), then it is both left anodyne and right anodyne. That is, we have inclusions
All of these inclusions are strict (see Example 220.127.116.11).