equivalences in/of $(\infty,1)$-categories
The general notion of opposite (∞,1)-category leads to a notion of opposite of a quasi-category , when (∞,1)-categories are incarnated as quasi-categories.
So the notion of opposite of a quasi-category generalizes the notion of opposite category from category theory.
Under the relation between quasi-categories and simplicial categories the opposite quasi-category is that corresponding to the obvious opposite SSet-enriched category. Concretely in terms of the simplicial set $S$ underlying the quasi-category, this amounts to reversing the order of the face and degenracy maps:
Secton 1.2.1 in