\oplus V^{\otimes  2}  \oplus  V^{\otimes  3}  \oplus \dots\,.$$

For an operad ${{\cal O}}$ (resp. a cooperad ${{\cal C}}$) and a cochain complex $V$ we denote by ${{\cal O}}(V)$ (resp. ${{\cal C}}(V)$) the free ${{\cal O}}$-algebra generated by $V$ (resp. the cofree ${{\cal C}}$-coalgebra cogenerated by $V$). The notation ${{\rm Cobar}}({{\cal C}})$ is reserved for the cobar construction [@Fresse], [@GJ] of a cooperad ${{\cal C}}$.

By “suspension” of a (co)operad ${{\cal O}}$ we mean the (co)operad ${{\Lambda}}({{\cal O}})$ whose $m$-th space is $$\label{susp-op}
{{\Lambda}}({{\cal O}})(m) = {{\bf s}}^{1-m} {{\cal O}}(m) \otimes {{\rm s g n}}_{m}\,,$$ where ${{\rm s g n}}_{m}$ is the sign representation of the symmetric group $S_m$. For example, a ${{\Lambda}}{{\bf Lie}}$-algebra is nothing but a Lie algebra with the bracket of degree $-1$. Similarly ${{\Lambda}}{{\bf coassoc}}$ is the cooperad governing coassociative coalgebras with comultiplication of degree $-1$.

It is not hard to see that for every (co)operad ${{\cal O}}$ and for every graded vector space $V$ we have the isomorphism $$\label{La-O-V}
{{\Lambda}}({{\cal O}}) (V) = {{\bf s}}\, {{\cal O}}({{\bf s}}^{-1} V)\,.$$

It is important to make a remark about degrees of cooperations in coalgebras over cooperads. If ${{\cal C}}$ is a cooperad and $v$ is a homogeneous vector 