Sheaves form topos

We know from Proposition A.15 that the inclusion admits an accessible left adjoint \(L\colon \PSh(C) \to \Shv_{\tau}(C)\). We need to show that \(L\) is left exact. By Corollary 4.16, it suffices to show that the class of morphisms \(K\) inverted by \(L\) is a congruence.The class \(K\) is precisely the strong saturation \(\tau^{ss}\) of \(\tau\). We claim that it agrees with \(\tau^c\). Since \(\tau\) consists of monomorphisms, it follows from Corollary 5.43 that \(\tau^c = \tau^m\). Choose the representable presheaves as a set of generators of \(\PSh(C)\). In the notation of Proposition 5.32, the set \(\tau^{bc}\) then consists of the pullbacks of the sieves in \(\tau\) along maps from representable presheaves. It agrees with \(\tau\): one inclusion follows from pullback stability, while the other follows by taking the identity map of the codomain of each sieve. Hence Proposition 5.32 gives \(\tau^m = (\tau^{bc})^s = \tau^s\). All in all, we see that \(\tau^{ss} \subseteq \tau^c = \tau^m = \tau^s \subseteq \tau^{ss}\), and thus all four classes agree. In particular \(\tau^{ss}\) is a congruence, finishing the proof.