nLab
finitely generated object

An object A of a concrete category C is finitely generated if it is a quotient object of some free object F in C, where F is free on a finite set. (We probably want ‘quotient object’ here to be interpreted in the sense of a regular epimorphism from F to C, although perhaps we explicitly want it to be the quotient of a congruence on F.)

The object A is finitely presented if it is the quotient of a congruence RF such that R is also free on a finite set. (Or maybe we don't demand that this is a congruence but accept any coequaliser of this form?)

See discussion at finitely presentable object more a more abstract version of these.