Background
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In a quasi-category the notion of terminal object known from ordinary category theory is relaxed in the homotopy theoretic sense to the suitable notion in (∞,1)-category theory:
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.
Let be a quasi-category and one of its objects (a vertex in the corresponding simplicial set). The object is a terminal object in if the following equivalent conditions hold:
The projection from the over quasi-category is a trivial fibration of simplicial sets.
For every object of the right hom Kan-complex into is contractible:
A quick survey is on page 159 of
For more details see definition 1.2.12.3, p. 46 in
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