In category theory texts categories are referred to by terms like a) "let C be a category with such-and-such (inner) properties" b) "the category of [some object type + morphisms, e.g. groups or topological spaces]" (always (?) concrete categories) and c) "[some structure, e.g. a poset, a group] treated as a category" (mostly (?) abstract categories) What I wonder about is if and how a category C can be explicitly given without referring to a "standard model" as in b) (according to model theory where a theory is firstly given and models are searched for only after that). I.e. I was looking for the term "let C be *the* category with such-and-such (inner) properties" but didn't find it. If this can be achieved, the search for "models" seems natural, but what's the common name for a "model of a category"? Google knows little about "model of a category": 5 hits in conjunction with "category theory", 8 hits in conjunction with "morphism", 7 hits in conjunction with "functor". Thanks in advance! [For admin and other information see: http://www.mta.ca/~cat-dist/ ]