Lemma 15.2.19. Let \(C\) and \(D\) be \(\infty \)-categories with finite coproducts and let \(F\colon C \to D\) be a functor. Then \(F\) preserves finite coproducts if and only if the induced map \(\Span (F)\colon \Span _{\ct ,\all }(\Fin (C)) \to \Span _{\ct ,\all }(\Fin (D))\) preserves cocartesian morphisms over \(\Span (\Fin )\).
Proof. The map \(\Span (F)\) is always a morphism of \(\infty \)-operads, so it remains to check cocartesian morphisms over forward spans. Under the unfurling description from Proposition 15.2.1, Proposition 15.2.8, their restrictions along \(\Fin \hookrightarrow \Span (\Fin )\) are the cocartesian morphisms in \(\Fin (C)\) and \(\Fin (D)\) which form coproducts over the fibers of a map of finite sets. The functor \(\Fin (F)\) preserves these morphisms if and only if \(F\) preserves finite coproducts. □
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