instead of demanding that from any other object there is a unique morphism into the terminal object, in a quasi-category there is a contractible space of such morphisms, i.e. the morphism to the terminal object is unique up to homotopy.

Definition

Let $C$ be a quasi-category and $c \in C$ one of its objects (a vertex in the corresponding simplicial set). The object $c$ is a terminal object in $C$ if the following equivalent conditions hold: