in ${{\cal C}}(m)$ then the corresponding cooperation $$\label{map}
A \to A^{\otimes \, m}$$ on a ${{\cal C}}$-coalgebra $A$ has the degree $-|v|$. In this paper by degree of a cooperation we mean the degree of the corresponding vector in the cooperad but not the degree of the map. For example, a vector ${{\Delta}}\in {{\Lambda}}{{\bf coassoc}}(2)$ has the degree $-1$ but the corresponding map $${{\Delta}}: A  \to  A \otimes A$$ for a ${{\Lambda}}{{\bf coassoc}}$-coalgebra $A$ has the degree $1$.

In this paper we often encounter $L_{\infty}$-algebras (resp. (DG) Lie algebras) with the binary operation having degree $-1$. Following our notation we call them ${{\Lambda}}{{\bf Lie}}_{\infty}$-algebras (resp. ${{\Lambda}}{{\bf Lie}}$-algebras). However, we still retain the word “$L_{\infty}$-morphism” for a morphism in the category of ${{\Lambda}}{{\bf Lie}}_{\infty}$-algebras.

Thus a ${{\Lambda}}{{\bf Lie}}_{\infty}$-structure on a graded vector space ${{\cal L}}$ is a degree $1$ codifferential ${\hat{Q}}_{{{\cal L}}}$ on the cofree ${{\Lambda}}^2{{\bf cocomm}}$ coalgebra $${{\Lambda}}^2{{\bf cocomm}}({{\cal L}})  = {{\bf s}}^2 S ({{\bf s}}^{-2} {{\cal L}})\,.$$ Similarly, an $L_{\infty}$-morphism ${\hat{U}}$ from a ${{\Lambda}}{{\bf Lie}}_{\infty}$-algebra ${{\cal L}}$ to a ${{\Lambda}}{{\bf Lie}}_{\infty}$-algebra ${{\widetilde{\mathcal L}}}$ is a morphism of ${{\Lambda}}^2{{\bf cocomm}}$-coalgebras $${\hat{U}}:  {{\bf s}}^2 S ({{\bf s}}^{-2} {{\cal L}})  \to  {{\bf s}}^2 S ({{\bf s}}^{-2} {{\widetilde{\mathcal L}}})$$ compatible with the codifferentials.

