Some remarks on autoequivalences of categories

Prof. Boris I. Plotkin [arXiv:math.GM/0210194, arXiv:math.GM/0204245] drew attention to the question when an equivalence between two categories is isomorphic as a functor to an isomorphism between them. It turns out that it is quite important for universal algebraical geometry and concerns mainly the categories $\mathrm{\Theta }\text{sp}0(X)$ of free universal algebras of some variety $\mathrm{\Theta }$ free generated by finite subsets of $X$. In the paper, a complete answer to the Plotkin’s question is given: there are no proper autoequivalences of the category $\mathrm{\Theta }\text{sp}0(X)$. Also some connected problems are discussed.