Lemma 6.150.
The functor \(\Cat^{\mathrm{ca}} \to \Topos\) given by \(C \mapsto \Fun^{\omega}(C,\An)\) is a fully faithful \(2\)-functor.
Proof
By Corollary 6.141, the functor \(\Pt\colon \Topos \to \Cat^{\acc,\omega}\) admits a left adjoint given by \(C \mapsto \Fun^{\omega}(C,\An)\). The unit of this adjunction provides a map \(C \to \Pt(\Fun^{\omega}(C,\An))\), which by Remark 6.149 is an equivalence when \(C\) is compactly assembled. Therefore the restriction of this adjunction to \(\Cat^{\mathrm{ca}}\) exhibits \(\Cat^{\mathrm{ca}} \to \Topos\) as fully faithful.