Lemma 6.174.
Every topos is a locus. Conversely, a locus \(T\) is a topos if and only if it admits a strictly initial object.
Proof
It is clear that every topos is a locus admitting a strictly initial object. Conversely, if \(T\) is a locus admitting a strictly initial object, then it in particular admits colimits, so \(T\) is presentable. Moreover, the functor \(T\catop \to \Cat, X \mapsto T_{/X}\) preserves weakly contractible limits (\(T\) a locus) and the terminal object (as \(\emptyset \in T\) strictly initial, see Example 2.11), hence all limits.