A categorical approach to study model comparison games in terms of comonads was recently initiated by Abramsky et al. In this work, we analyse games that appear naturally in the context of description logics and knowledge representation. We consider expressive sublogics of ALCIOQbSelf, namely, the logics that extend ALC with any combination of inverses, nominals, number restrictions, safe boolean roles combinations and the Self operator. Our construction augments and modifies the so-called modal comonad by Abramsky and Shah. The approach that we took heavily relies on the use of relative monads, which we leverage to encapsulate additional capabilities within the bisimilation games in a compositional manner.