Example 2.11. (Initial object)
Let \(T\) be a category with an initial object. Then the initial object \(\emptyset\) is van Kampen if and only if it is strictly initial, meaning that any morphism \(X \to \emptyset\) is an isomorphism.
Indeed, by definition \(\emptyset\) is van Kampen 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 universality condition.