Let be a locally small category that admits filtered colimits of monomorphisms. Then an object is finitely generated if the corepresentable functor
preserves these filtered colimits of monomorphisms. This means that for every filtered category and every functor such that is a monomorphism for each morphism of , the canonical morphism
is an isomorphism.
An object of a concrete category is finitely generated if it is a quotient object (in the sense of a regular epimorphism) of some free object in , where is free on a finite set.
The object is finitely presented if it is the coequalizer of a parallel pair such that is also free on a finite set.
A set is a finitely generated object in Set iff it is (Kuratowski-)finite. For this to hold constructively, filtered categories (appearing in the definition of filtered colimit) have to be understood as categories admitting cocones of every Bishop-finite diagram.
For a ring, an -module is finitely generated if it is a quotient of a free module with a finite basis.
The general definition is in Locally Presentable and Accessible Categories, definition 1.67.