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".)