The notion of "model of a category"

The notion of "model of a category"

[Hans-Peter Stricker]

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!



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