Every logos admits a presentation

We saw in the proof of Theorem 2.42 that there is a small category \(C\) with finite limits and an accessible left exact localization Let \(i\colon C\hookrightarrow C^{\lex}\) be the canonical functor, and let \(F\colon C^{\lex}\to C\) be the left exact extension of \(\id_C\). The universal property of \(C^{\lex}\) gives for \(X\in C\) and \(Y\in C^{\lex}\). Thus \(i\dashv F\), and the unit \(\id_C\to Fi\) is an isomorphism.The left Kan extension along \(F\) is therefore restriction along \(i\): In particular, \(F_!\) is left exact. Its right adjoint is the restriction functor \(F^*\colon\PSh(C)\to\PSh(C^{\lex})\), which is fully faithful because \(Fi\simeq\id_C\). Hence \(F_!\) is a left exact Bousfield localization. It follows that the composite is again a left exact Bousfield localization.Its kernel \(K:=\ker(Q)\) is a congruence of small generation by Corollary 4.16. Choose a small class \(\Sigma\subseteq K\) which generates \(K\) as a strongly saturated class. Since \(K\) is a congruence, \(\Sigma^c\subseteq K\); conversely, \(\Sigma^c\) is strongly saturated and contains \(\Sigma\), so \(K\subseteq\Sigma^c\). Hence \(K=\Sigma^c\). The universal property of the quotient now identifies which is the desired presentation.