equivalences in/of $(\infty,1)$-categories
The notion of inner fibration of simplicial sets is one of the notion of fibrations of quasi-categories.
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 $C \to D$ such that each fiber is a quasi-category and such that over each morphism $f : d_1 \to d_2$ of $D$, $C$ may be thought of as the cograph of an (∞,1)-profunctor $C_{d_1} ⇸ C_{d_2}$.
So when $D = {*}$ is the point, an inner fibration $C \to {*}$ is precisely a quasi-category $C$.
And when $D = N(\Delta[1])$ is the nerve of the interval category, an inner fibration $C \to \Delta[1]$ may be thought of as the cograph of an (∞,1)-profunctor $C ⇸ D$.
This $(\infty,1)$-profunctor comes form an ordinary (∞,1)-functor $F : C \to D$ precisely if the inner fibration $K \to \Delta[1]$ is even a coCartesian fibration. And it comes from a functor $G : D \to C$ 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 $(\infty,1)$-profunctor $K \to \Delta[1]$ is both a Cartesian as well as a coCartesian fibration does it exhibit a pair of adjoint (∞,1)-functors.
A morphism of simplicial sets $f : X \to S$ is an inner fibration or inner Kan fibration if its has the right lifting property with respect to all inner horn inclusions, i.e. if for all commuting diagrams
for $0 \lt i \lt n$ 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 $f : X \to Y$ of simplicial sets may be factored as
with $X \to Z$ a left/right/inner anodyne cofibration and $Z \to Y$ accordingly a left/right/inner Kan fibration.
inner 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.