When the notion of (∞,1)-category is incarnated in terms of the notion of quasi-category, an inner fibration is a morphism of simplicial sets such that each fiber is a quasi-category and such that over each morphism of , may be thought of as the cograph of an (∞,1)-profunctor .
So when is the point, an inner fibration is precisely a quasi-category .
This -profunctor comes form an ordinary (∞,1)-functor precisely if the inner fibration is even a coCartesian fibration. And it comes from a functor precisely if the fibration is even a Cartesian fibration. This is the content of the (∞,1)-Grothendieck construction.
And precisely if the inner fibration/cograph of an -profunctor is both a Cartesian as well as a coCartesian fibration does it exhibit a pair of adjoint (∞,1)-functors.
for there exists a left
The morphisms with the left lifting property against all inner fibrations are called inner anodyne.
By the small object argument we have that every morphism of simplicial sets may be factored as
with a left/right/inner anodyne cofibration and accordingly a left/right/inner Kan fibration.
Inner fibrations were introduced by Andre Joyal. A comprehensive account is in section 2.3 of
Their relation to cographs/correspondence is discussed in section 2.3.1 there.