Category theorists are mostly conceptual mathematicians that use category theory as their main tool of choice; it is less common to meet category theorists that are interested in categories in the sense that a group theorist is interested in a group.
If this is what we want to transmit, the name “category theorist” may not be the best: we work with categories, true, but we also work with bicategories, premonoidal categories or profunctors. If we want to say that these are all concepts derived from a more primitive one, that one is possibly “monoid”, more than “category”.
Monoids are the simpler algebra of composition, category theorists study composition; we should probably be calling them “monoid theorists”. If we want to convey we study categories, we would not say that we are “bicategory theorists”, we would reserve that term for the specialists in the theory of bicategories.
Tags: opinion.