Lemma 14.2.4. Let \(p_{\Oo }\colon \Oo ^{\otimes } \to \Span (\Fin )\) be a cocartesian fibration. Then \((\Oo ^{\otimes },p_{\Oo })\) is an \(\infty \)-operad if and only if the cocartesian straightening of \(p_{\Oo }\) \[ \Str ^{\cc }(p_{\Oo })\colon \Span (\Fin ) \to \Cat _{\infty } \] preserves finite products.
Proof. We check conditions (i)–(iii) of Proposition 14.1.9. Condition (i) is part of the assumption on \(p_{\Oo }\). Since the spans \(\rho _i\colon I \hookleftarrow \{i\} \xrightarrow {=} \{i\}\) exhibit \(I\) as the product of the one-element sets in \(\Span (\Fin )\), condition (ii) is equivalent to preservation of finite products by \(\Str ^{\cc }(p_{\Oo })\). Under this assumption, Reference ? of [Cisinski et al. (2026)] shows that the cocartesian lifts of the maps \(\rho _i\) exhibit every object over \(I\) as the product of its components, which is condition (iii). □
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