Strictly initial objects and descent

Example 23.7.8 (Strictly initial object). Let \(T\) be an \(\infty \)-category with an initial object. Then the initial object \(\emptyset \) satisfies descent if and only if it is strictly initial, meaning that any morphism \(X \to \emptyset \) is an isomorphism.

Indeed, by definition \(\emptyset \) satisfies descent if and only if the functor \(T_{/\emptyset } \to *\) is an equivalence. This functor admits a fully faithful left adjoint \(* \to T_{/\emptyset }\) hitting the identity \(\id _{\emptyset }\). For the functor to be an equivalence, the counit \(\emptyset \to X\) must be an isomorphism for every \(X \in T_{/\emptyset }\), which is precisely the strict initiality condition.