This follows by elimination of imaginaries in algebraically closed fields. Given a finite affine cover of your scheme, one obtains a functor to Set by gluing the affine charts. This is indeed not definable, but a quotient of a definable set by a definable equivalence relation. Elimination of imaginaries tells you precisely that such a quotient is in definable bijection with a definable set. You can find the details in Chapter 4 (A. Pillay, Model theory of algebraically closed fields) of the following book:
Bouscaren, Elisabeth, Introduction to model theory, Bouscaren, Elisabeth (ed.), Model theory and algebraic geometry. An introduction to E. Hrushovski’s proof of the geometric Mordell-Lang conjecture. Berlin: Springer. Lect. Notes Math. 1696, 1-18 (1998). ZBL0925.03160.
Chapters 1 and 2 also contain an introduction to general model theoretic concepts including elimination of imaginaries (Chapter 2).
