# Group objects via diagrams or generalized elements — Kripke–Joyal?

The notion of a group object $$G$$ in a category with finite products can either be defined with a few commutative diagrams or via requiring that each hom set $$\hom(X,G)$$ is a group. There is a theorem that these two definitions are equivalent.

Question: Does the equivalence of these two definitions follow from Kripke–Joyal semantics (Theorem 3.2)?

After all, Kripke–Joyal semantics gives a characterization of truth in the internal language of a topos using generalized elements. (And the second definitions renders as "for each $$X$$, the generalized elements with stage $$X$$ constitute a group".)