In our recent paper [Sh1] a version of the generalized Deligne conjecture for abelian $n$-fold monoidal categories is proven. For $n=1$ this result says that, given an abelian monoidal $k$-linear category $mathscr{A}$ with unit $e$, $k$ a field of characteristic 0, the dg vector space $mathrm{RHom}_{mathscr{A}}(e,e)$ is the first component of a Leinster 1-monoid in $mathscr{A}lg(k)$ (provided a rather mild condition on the monoidal and the abelian structures in $mathscr{A}$, called homotopy compatibility, is fulfilled). In the present paper, we introduce a new concept of a ${it graded}$ Leinster monoid. We show that the Leinster monoid in $mathscr{A}lg(k)$, constructed by a monoidal $k$-linear abelian category in [Sh1], is graded. We construct a functor, assigning an algebra over the chain operad $C(E_2,k)$, to a graded Leinster 1-monoid in $mathscr{A}lg(k)$, which respects the weak equivalences. Consequently, this paper together with loc.cit. provides a complete proof of the generalized Deligne conjecture for 1-monoidal abelian categories, in the form most accessible for applications to deformation theory (such as Tamarkins proof of the Kontsevich formality).
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