Segal condition for finite pointed sets

Definition 5.3.7 (Segal condition). Let \(C\) be an \(\infty \)-category with finite products. Denote by \(\Fin _*\) the category of finite pointed sets, whose objects are pairs \((S,*)\) consisting of a finite set \(S\) and a distinguished basepoint \(* \in S\), and whose morphisms are basepoint-preserving maps. For a finite set \(S\), we write \(S_+ := S \sqcup \{*\}\) for \(S\) with an added disjoint basepoint.

A functor \(M\colon \Fin _* \to C\) is said to satisfy the Segal condition if for every finite set \(S\) the canonical map \[ M(S_+) \to \prod _{s \in S} M(\{s\}_+) \] induced by the pointed maps \(S_+ \to \{s\}_+\) is an isomorphism, where the pointed map sends everything except for \(s \in S\) to the base point of \(\{s\}_+\). We denote by \[ \Fun ^{\Segal }(\Fin _*,C) \subseteq \Fun (\Fin _*,C) \] the full subcategory spanned by functors satisfying the Segal condition.