Let \({\mathcal {V}}\) be a variety of algebras of some type \(\Omega \). An interest to describing automorphisms of the category \(\Theta ^0 ({\mathcal {V}})\) of finitely generated free \({\mathcal {V}}\)-algebras was inspired by development of universal algebraic geometry founded by B. Plotkin. There are a lot of results on this subject. A common method of getting such results was suggested and applied by B. Plotkin and the author. The method is to find all terms in the language of a given variety which determine such \(\Omega \)-algebras that are isomorphic to a given \(\Theta ^0 ({\mathcal {V}})\)-algebra and have the same underlying set with it. But this method can be applied only to automorphisms which take all objects to isomorphic ones. The aim of the present paper is to suggest another method which works in more general setting. This method is based on two main theorems. The first of them gives a general description of automorphisms of categories which are supplied with a faithful representative functor into the category of sets. The second one shows how to obtain the full description of automorphisms of the category \(\Theta ^0 ({\mathcal {V}})\). This part of the paper ends with two examples. The first of them shows the preference of our method in a known situation (the variety of all semigroups) and the second one demonstrates obtaining new results (the variety of all modules over arbitrary ring with unit). The last section contains some applications to universal algebraic geometry.

令\({\mathcal {V}}\)为某种类型\(\Omega \)的各种代数。对描述有限生成的自由\({\mathcal {V}}\) -代数范畴\(\Theta ^0 ({\mathcal {V}})\)自同构的兴趣受到通用代数几何发展的启发由 B. Plotkin 创立。关于这个主题有很多结果。B. Plotkin 和作者提出并应用了获得此类结果的常用方法。该方法是找到给定变体语言中的所有项，这些项确定与给定\(\Theta ^0 ({\mathcal {V}})\) -代数同构的\(\Omega \) -代数并与其具有相同的基础集。但这种方法只能应用于使所有对象同构的自同构。本文的目的是提出另一种适用于更一般环境的方法。该方法基于两个主要定理。第一个给出了范畴自同构的一般描述，范畴自同构被提供给集合范畴的忠实代表函子。第二个展示了如何获得类别\(\Theta ^0 ({\mathcal {V}})\)的自同构的完整描述。本文的这一部分以两个示例结束。第一个展示了我们的方法在已知情况下的偏好（所有半群的多样性），第二个展示了获得新结果（所有模相对于带单位的任意环的多样性）。最后一部分包含通用代数几何的一些应用。