Span patterns are weakly extensive

Remark 17.3.6. If \(F_L\) contains the fold maps \(I \sqcup I \to I\), then the conditions of Convention 17.3.1 imply that \((F,F_L,F_R)\) is weakly extensive. Indeed, weak coextensivity implies that \(F_L\) and \(F_R\) are closed under finite coproducts, while part (3) of Convention 17.3.1 supplies the morphisms \(\emptyset \to I\) in \(F_L\). The claim therefore follows from part (1) of Lemma 13.3.4. It follows that \(F_L\) admits finite coproducts, and a functor \(F_L\catop \to C\) satisfies the Segal condition if and only if it preserves finite products. Similarly, part (1) of Lemma 13.3.8 shows that \(\Span _{L,R}(F)\) admits finite products, and a functor \(M\colon \Span _{L,R}(F) \to C\) is an \(\Ff \)-monoid if and only if it preserves finite products.