In monoidal abelian categories with enough right-flat projectives, the co-Hochschild complex of the unit's projective resolution carries a B_infinity-structure that is A_infinity-quasi-isomorphic to the derived endomorphism algebra of the unit and recovers the Hochschild complex for bimodules.
Hermann, Monoidal categories and the Gerstenhaber bracket in Hochsc hild cohomology , Mem
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The $B_\infty$-structure on the derived endomorphism algebra of the unit in a monoidal category
In monoidal abelian categories with enough right-flat projectives, the co-Hochschild complex of the unit's projective resolution carries a B_infinity-structure that is A_infinity-quasi-isomorphic to the derived endomorphism algebra of the unit and recovers the Hochschild complex for bimodules.