A \(\mathbb{T}_2\)-algebra in a unital operad \(\mathcal O\) is given by a pair of colors \(X,Y \in \underline{\mathcal O}\) equipped with an element of the pullback \[\mathrm{Mul}_{\mathcal O}(X,X;Y) \times_{\mathrm{Hom}_{\underline{\mathcal O}}(X,Y)^{\times 2}} \mathrm{Hom}_{\underline{\mathcal O}}(X,Y),\] i.e. a \(2\)-ary operation \(\mu \in \mathrm{Mul}_{\mathcal O}(X,X;Y)\) and an identification of the \(1\)-ary operations \(\mu(-, 1)\) and \(\mu(1,-)\) in \(\mathrm{Hom}_{\underline{\mathcal O}}(X,Y)\).
A \(\mathbb T_2\)-algebra in a (not necessarily unital) \(\infty\)-operad \(\mathcal O\) is a \(\mathbb T_2\)-algebra in the unital \(\infty\)-operad \(\mathrm{Alg}_{\mathbb E_0}(\mathcal O)\).