ScalingStacks

In particular, it follows that the unit of the adjunction is an operad map \(\mathcal O\rightarrow\underline{\mathcal O}_{\sqcup}\) which induces the identity on underlying \(\infty\)-categories \(\underline{\mathcal O} \rightarrow\underline{\underline{\mathcal O}_{\sqcup}} \simeq \underline{\mathcal O}\). Fixing colors \(X_1, \ldots, X_n, Y \in \underline{\mathcal O}\), this induces a map \[ \sigma \colon \mathrm{Mul}_{\mathcal O}(X_1, \ldots, X_n; Y) \rightarrow\mathrm{Hom}_{\underline{\mathcal O}}(X_1, Y) \times \cdots \times \mathrm{Hom}_{\underline{\mathcal O}}(X_n, Y).\] The components \(\mathrm{Mul}_{\mathcal O}(X_1, \ldots, X_n; Y) \rightarrow\mathrm{Hom}_{\underline{\mathcal O}}(X_i, Y)\) of this map may be thought of as inserting units in all but the \(i\)-th slot.