Objects as functors from the terminal category

Remark 1.3.2. We further assume that the canonical functor \([0] \to *\) is an equivalence. In particular, objects of an \(\infty \)-category \(C\) may equally well be regarded as functors \([0] \to C\) or as functors \(* \to C\).

In these empty-index cases we also silently assume one extra layer of coherence: any two natural isomorphisms between two functors \(T \to *\) are connected by a 3-isomorphism, and similarly for functors \(\emptyset \to T\). This mild strengthening is only used when comparing products with pullbacks over \(*\) and coproducts with pushouts under \(\emptyset \), for instance in Exercise 1.3.10.