  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 supplement them with
suitable game comonads. More precisely, we consider expressive sublogics of
ALCSelfIbO, namely, the logics that extend ALC with any combination of
inverses, nominals, safe boolean roles combinations, and 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 comonads,
which we leverage to encapsulate additional capabilities within the
bisimulation games in a compositional manner.
