Lemma 17.3.17. Let \(\Oo \) be an \(\infty \)-operad, and let \(\{x_i\}_{i \in I}\) and \(\{y_j\}_{j \in J}\) be objects of \(\Env (\Oo )\). Then there is a natural equivalence \[ \Hom _{\Env (\Oo )}(\{x_i\}_{i \in I}, \{y_j\}_{j \in J}) \simeq \coprod _{f\colon I \to J} \; \prod _{j \in J} \Oo (\{x_i\}_{i \in f^{-1}(j)}; y_j), \] where the coproduct ranges over all maps of finite sets \(f\colon I \to J\). In particular, for a single output color \(y \in \Oo ^{\simeq }\) there is a natural equivalence \[ \Hom _{\Env (\Oo )}(\{x_i\}_{i \in I}, y) \simeq \Oo (\{x_i\}_{i \in I}; y). \]
Proof. As noted above, the underlying \(\infty \)-category of \(\Env (\Oo )\) is the pullback \(\Fin \times _{\Span (\Fin )} \Oo ^{\otimes }\) along the inclusion \(\Fin \hookrightarrow \Span (\Fin )\) of the forward spans. Consequently, \(\Hom _{\Env (\Oo )}(\{x_i\}_{i \in I}, \{y_j\}_{j \in J})\) is the pullback of \[ \Hom _{\Oo ^{\otimes }}(\{x_i\}_{i \in I}, \{y_j\}_{j \in J}) \longrightarrow \Hom _{\Span (\Fin )}(I,J) \] and the subanima \(\Hom _{\Fin }(I,J)\) of forward spans, which produces the coproduct over the maps \(f\colon I \to J\). Fixing such an \(f\), condition (3) of Definition 17.3.10 identifies the fiber over \(f\) with the product over \(j \in J\) of the fibers of \(\Hom _{\Oo ^{\otimes }}(\{x_i\}_{i \in I}, y_j)\) over the span \(I \hookleftarrow f^{-1}(j) \to \lra {1}\). Cocartesian transport along the inert morphism \(\{x_i\}_{i \in I} \to \{x_i\}_{i \in f^{-1}(j)}\) identifies the latter fiber with the multimorphism anima \(\Oo (\{x_i\}_{i \in f^{-1}(j)}; y_j)\). The final statement is the case \(J = \lra {1}\). □
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