The syntactic category of the theory of groups is the “general abstract” of groups. Its essentially unique generating object is the abstract particular group.
The category Grp of all groups is the concrete general of groups.
An object in there is some group: a concrete particular.
The category-theoretic formalization of these notions as proposed by Bill Lawvere is disussed in print for instance in
Bill Lawvere, Categorical refinement of a Hegelian principle, section 1 of Bill Lawvere, Tools for the advancement of objective Logic: Closed categories and toposes, in John Macnamara, Gonzalo Reyes, the logical foundations of cognition, Oxford University Press (1994)