If \(\mathcal P\) is an operad and \(\mathcal C\) a cooperad and \(\alpha:\mathcal C\longrightarrow \mathcal P\) an operadic twisting, then of \(A\) an algebra on \(\mathcal P\) and \(C\) a coalgebra on \(\mathcal C\), the chain complex of linear maps from \(C\) to \(A\) is canonically a homotopy Lie algebra (or \(L_\infty\)-algebra), denoted by \(\mathrm{Hom}^\alpha(C,A)\).
The first main result of this paper is the following: for any pair of algebras on \(\mathcal P\) and any pair of conilpotent coalgebras on \(\mathcal C\), a \(L_\infty\)-algebra is defined, whose Maurer-Cartan elements are in natural bijection with the \(\infty\)-morphisms between the algebras, respectively the coalgebras. Moreover, the gauge equivalence for Maurer-Cartan elements corresponds to the homotopy for morphisms.
The second main result is the following: if \(\alpha\) is Koszul, the bifunctor \(\mathrm{Hom}^\alpha(-,-)\) extends to \(\infty\)-morphisms in both slots, if one accepts to work up to homotopy.