  It has been proven by Schupp and Bergman that the inner automorphisms of
groups can be characterized purely categorically as those group automorphisms
that can be coherently extended along any outgoing homomorphism. One is thus
motivated to define a notion of (categorical) inner automorphism in an
arbitrary category, as an automorphism that can be coherently extended along
any outgoing morphism, and the theory of such automorphisms forms part of the
theory of covariant isotropy. In this paper, we prove that the categorical
inner automorphisms in any category $\mathsf{Group}^{\mathcal{J}}$ of
presheaves of groups can be characterized in terms of conjugation-theoretic
inner automorphisms of the component groups, together with a natural
automorphism of the identity functor on the index category $\mathcal{J}$. In
fact, we deduce such a characterization from a much more general result
characterizing the categorical inner automorphisms in any category
$\mathbb{T}\mathsf{mod}^{\mathcal{J}}$ of presheaves of $\mathbb{T}$-models for
a suitable first-order theory $\mathbb{T}$.
