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