  An important result in quasi-category theory due to Lurie is the that
cocartesian fibrations are exponentiable, in the sense that pullback along a
cocartesian fibration admits a right Quillen right adjoint that moreover
preserves cartesian fibrations; the same is true with the cartesian and
cocartesian fibrations interchanged. To explicate this classical result, we
prove that the pullback along a cocartesian fibration between quasi-categories
forms the oplax colimit of its "straightening," a homotopy coherent diagram
valued in quasi-categories, recovering a result first observed by Gepner,
Haugseng, and Nikolaus. As an application of the exponentiation operation of a
cartesian fibration by a cocartesian one, we use the Yoneda lemma to construct
left and right adjoints to the forgetful functor that carries a cartesian
fibration over B to its obB-indexed family of fibers, and prove that this
forgetful functor is monadic and comonadic. This monadicity is then applied to
construct the reflection of a cartesian fibration into a groupoidal cartesian
fibration, whose fibers are Kan complexes rather than quasi-categories.
