This is in continuation of the previous post. I know what a compact topos: $F$ in $Set$ is:
$F$ is compact if the geometric morphism $\gamma: F \rightarrow Set$ preserves directed suprema of subojects of $1$.
I am looking at the paper Moerdijk, Vermeulen pg 6
Def 1.3. A map $f: F \rightarrow E$ of topos is proper if it renders $F$ compact as an $E$ topos.
What does this mean precisely?
Presumably it means that $f$ preserves directed suprema subobjects of $1$. However at least for my topos-theoretic level the definition seems a bit ambiguous-I’m not sure whether some aspect of this should be interpreted internally to $E$. An $E$-topos is just a topos with a map to $E$, a generalization of the way in which every cocomplete topos has a canonical map to Set.