  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 Tamarkin's
proof of the Kontsevich formality).
