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Let $A=A(x_{1},...,x_{n})$ be a free associative algebra in $\mathcal{A}$freely generated over $K$ by a set $X=\{x_{1},...,x_{n}\}$, $End A$ be thesemigroup of endomorphisms of $A$, and $Aut End A$ be the group ofautomorphisms of the semigroup $End A$. We investigate the structure of thegroups $Aut End A$ and $Aut \mathcal{A}^{\circ}$, where $\mathcal{A}^{\circ}$is the category of finitely generated free algebras from $\mathcal{A}$. Weprove that the group $Aut End A$ is generated by semi-inner and mirrorautomorphisms of $End F$ and the group $Aut \mathcal{A}^{\circ}$ is generatedby semi-inner and mirror automorphisms of the category $\mathcal{A}^{\circ}$.  This result solves an open Problem formulated in \cite{22

AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE ASSOCIATIVE ALGEBRA