lax idempotent 2-monad
Let F:S→CF:S\to C be a functor between cartesian categories, having a right adjoint RR. Then RR has a canonical extension to an SS-indexed functor ℂ→𝕊\mathbb{C}\to \mathbb{S}, where 𝕊:S op→Cat\mathbb{S}:S^{op}\to Cat, X↦S/XX\mapsto S/X, f↦f *f\mapsto f^* (pullback along ff) is self-indexing of SS and C I:=C/F(I)C^I:=C/F(I).
This is also in indexed category?.