We prove that every automorphism of the category of free Lie algebras is a semi-inner automorphism. This solves Problem 3.9 from [G. Mashevitzky, B. Plotkin, E. Plotkin, Electron. Res.

Abstract. Let Θ be an arbitrary variety of algebras and let Θ 0 be the category of all free finitely generated algebras from Θ. We study automorphisms of such categories for special Θ. The cases of the varieties of all groups, all semigroups, all modules over a noetherian ring, all associative and commutative algebras over a field are completely investigated. The cases of associative and Lie algebras are also considered. This topic relates to algebraic geometry in arbitrary variety of algebras Θ.

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Abstract. We determine all isomorphisms between the endomorphism semigroups of free monoids or free semigroups and prove that automorphisms of the endomorphism semigroup of a free monoid or a free semigroup are inner or "mirror inner". In particular, we answer a question of B. I. Plotkin.

No abstract