Corollary 14.1.11. If \(\Oo \) is an \(\infty \)-operad, then the inert morphisms in \(\Oo ^{\otimes }\) are precisely the morphisms of the form \(\widetilde {f}\colon \prod _{i \in I} x_i \to \prod _{j \in J} x_{f(j)}\) for maps \(f\colon J \to I\) in \(\Fin \) and colors \(x_i \in \Oo ^{\simeq }\).
Similarly, if \(\Pp \) is another \(\infty \)-operad, and \(\phi ^{\otimes }\colon \Oo ^{\otimes } \to \Pp ^{\otimes }\) is a functor over \(\Span (\Fin )\), then \(\phi ^{\otimes }\) preserves finite products (i.e., corresponds to a morphism \(\phi \colon \Oo \to \Pp \) of \(\infty \)-operads) if and only if it preserves the inert morphisms.
Proof. By assumption \(\widetilde {f}\) is inert for every \(f\). Conversely, we saw in the proof of (i) of the previous proposition that for every backwards span \(I \xleftarrow {\smash {f}} J \xrightarrow {=} J\) and every \(X \in \Oo ^{\otimes }_I\) the map \(\widetilde {f} \colon X \simeq \prod _{i \in I} X_i \to \prod _{j \in J} X_{f(j)}\) is a cocartesian lift of \(f\) starting in \(X\), so by uniqueness it follows that every cocartesian lift is of this form.
If \(\phi ^{\otimes }\) preserves finite products, it in particular preserves the maps \(\widetilde {f}\colon \prod _{i \in I} X_i \to \prod _{j \in J} X_{f(j)}\), hence all inert morphisms. Conversely, if \(\phi ^{\otimes }\) preserves inert morphisms, it preserves the projection maps \(Y \to Y_j\) for \(Y \in \Oo ^{\otimes }_J\) exhibiting \(Y\) as a product of the \(Y_j\), hence it preserves every nonempty finite product. It also preserves the terminal object: since \(\phi ^{\otimes }\) lies over \(\Span (\Fin )\), it carries the unique object of \(\Oo ^{\otimes }_{\emptyset }\simeq *\) to the unique object of \(\Pp ^{\otimes }_{\emptyset }\simeq *\). □
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