{{\mathfrak{c}}}, \dots, {{\mathfrak{c}}}}_{k},  \underbrace{{{\mathfrak{o}}}, {{\mathfrak{o}}}, \dots, {{\mathfrak{o}}}}_{n}  ;  {{\mathfrak{c}}})\,,$$ $${{\cal O}}(\underbrace{{{\mathfrak{c}}}, {{\mathfrak{c}}}, \dots, {{\mathfrak{c}}}}_{k},  \underbrace{{{\mathfrak{o}}}, {{\mathfrak{o}}}, \dots, {{\mathfrak{o}}}}_{n}  ;  {{\mathfrak{o}}})$$ together with the right action of the group $S_k \times S_n$ and by the corresponding (co)operadic (co)multiplications for these objects.

We often use the following short-hand notation for objects of 2-colored (co)operads $$\label{notation-mc-ma}
{{\cal O}}^{{{\mathfrak{c}}}}(n,k) : = {{\cal O}}(\underbrace{{{\mathfrak{c}}}, {{\mathfrak{c}}}, \dots, {{\mathfrak{c}}}}_{n}, 
\underbrace{{{\mathfrak{o}}}, {{\mathfrak{o}}}, \dots, {{\mathfrak{o}}}}_{k}  ;  {{\mathfrak{c}}})\,,$$ $$\label{notation-mc-ma1}
{{\cal O}}^{{{\mathfrak{o}}}}(n,k) : = {{\cal O}}(\underbrace{{{\mathfrak{c}}}, {{\mathfrak{c}}}, \dots, {{\mathfrak{c}}}}_{n}, 
\underbrace{{{\mathfrak{o}}}, {{\mathfrak{o}}}, \dots, {{\mathfrak{o}}}}_{k}  ;  {{\mathfrak{o}}})\,.$$ Let us consider, for example, the operad ${{\bf Lie}}$ whose algebras are Lie algebras (in the category of vector spaces). This operad can be upgraded to the $2$-colored operad ${{\bf Lie}}^+$ whose algebras are pairs $({{\cal V}}, {{\cal W}})$ where ${{\cal V}}$ is a Lie algebra and ${{\cal W}}$ is a module over ${{\cal W}}$. To vectors in ${{\cal V}}$ (resp, ${{\cal W}}$) we assign the color ${{\mathfrak{c}}}$ (resp. ${{\mathfrak{o}}}$). It is not hard to see that for ${{\bf Lie}}^+$ we have $$({{\bf Lie}}^+)^{{{\mathfrak{c}}}}(n,0)= {{\bf Lie}}(n)\,,$$ $$({{\bf Lie}}^+)^{{{\mathfrak{o}}}}(n,1)= {{\mathbb K}}[S_n]\,,$$ and the remaining vector spaces are ${{\bf 0}}$.

\[rem:cofibrant\] For colored operads of cochain complexes we use a bit of terminology from closed model categories. Thus, 