The notion of -category combines that of a -category and of an -algebroid (i.e. a -linear category). Thus it might reasonably be called simply a -linear -category. However, -categories were popular in AQFT for a while, while the notion of dagger-category was introduced in the context of quantum information theory, so for historical reasons we have two names.
In a -category, the dagger operation is synonymously denoted .
See DHR category.