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 \(\textsf{Group}^\mathcal {J}\) 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 \(\mathbb {T}\textsf{mod}^\mathcal {J}\) of presheaves of \(\mathbb {T}\)-models for a suitable first-order theory \(\mathbb {T}\).

Schupp 和 Bergman 已经证明，群的内自同构可以纯粹分类地表征为那些可以沿任何输出同态相干扩展的群自同构。因此，人们有动机在任意范畴中定义一个（范畴的）内自同构的概念，作为可以沿着任何输出态射连贯地扩展的自同构，并且这种自同构的理论构成了协变各向同性理论的一部分。在本文中，我们证明了任何类别\(\textsf{Group}^\mathcal {J}\)群的预层可以根据组成群的共轭理论内自同构，连同索引范畴 \(\mathcal {J}\ )上的恒等函子的自然自同构来表征。事实上，我们从一个更普遍的结果中推导出这样的特征，该结果表征了 \( \ mathbb { T}\) - 适合一阶理论的模型\(\mathbb {T}\)。