Spahn notes on Lawvere-Tierney topologies, monads, object classifiers, local toposes, indexed functors

lax idempotent 2-monad

Lemma (Elephant B1.2 Lemma 1.2.3)

Let $F:S\to C$ be a functor between cartesian categories, having a right adjoint $R$. Then $R$ has a canonical extension to an $S$-indexed functor $\mathbb{C}\to \mathbb{S}$, where $\mathbb{S}:S^{op}\to Cat$, $X\mapsto S/X$, $f\mapsto f^*$ (pullback along $f$) is self-indexing of $S$ and $C^I:=C/F(I)$.

This is also in indexed category?.