$\newcommand{\Psh}{\operatorname{Psh}}
\newcommand{\Sh}{\operatorname{Sh}}
\newcommand{\O}{{\mathcal{O}}}$
Let $X$ be a locale, $\O(X)$ the corresponding frame.


1. What's the localic reflection of $\Psh X$?


We know that
$$
\O(X) \cong \mathrm{Sub}\_{\Sh X}(1)
$$
Call $Y = \mathrm{Sub}\_{\Psh X}(1)$ the localic reflection of $\Psh X$.
One has $\Psh X \simeq \Sh Y$.


Since $\Sh X$ is a subtopos of $\Psh X$, $X$ should be a sublocale of $Y$, i.e. there should be a nucleus $j$ on $\O(Y)$ such that $\O(X) = \O(Y) / j$.


2. Who's $j$?



What's the localic reflection of a presheaf topos?
I'm writing $\mathcal{O}(X)$ for the frame corresponding to $X$.


Opens of $Y$ are sieves on $X$, i.e. the collection of open subsets $V \subset \mathcal{O}(X)$ such that $v \in V$ and $u \leqslant v \Rightarrow u \in V$. Thus **$Y$ is the locale of downward closed subsets of $\mathcal O(X)$**.


More generally if $P$ is a poset, a subterminal presheaf on $P$ is a functor $P \to \{ 0,1\}$ and the set of $p \in P$ sent to $1$ is a sieve on $P$. This induces a bijection between sieves and subterminal presheaves.


When you identify $X$ as a sublocale of $Y$, its opens correspond to principal sieves, i.e. the ones of the form $\downarrow\! v = \{ u \in \mathcal{O}(X) | u \leqslant v \}$.


Thus the inclusion of locales $X \hookrightarrow Y$ is given by the quotient of frames $\mathcal{O}(Y) \to \mathcal{O}(X)$ defined as $V \mapsto \bigvee V$ and whose **right** adjoint is $v \mapsto {\downarrow\! v}$.


The nucleus $j$ hence takes a general sieve $V \subset \mathcal{O}(X)$ and sends it to the smallest principal ideal containing $V$, that is, $\downarrow\! \bigvee V$.


