fundamental theorem of topos theory

The “fundamental theorem” of topos theory (in the terminology of McLarty 1992) asserts that for any topos and any object, also the slice category is a topos: the slice topos. If is a category of sheaves, hence a Grothendieck topos, then so its its slice: (SGA4.1, p. 295). The analogous statement holds for slice -categories of -toposes: slice -toposes (Lurie 2009, Prop. 6.3.5.1). The archetypical special case is that slice categories of categories of presheaves over a representable are...