Lemma 4.25.
Let \((C,\Sigma)\) be a logos presentation. Then the quotient functor \(\An[C] \to \lra{C \mid \Sigma}\) is universal among logos morphisms \(\An[C] \to T\) that invert \(\Sigma\).
Proof
By Proposition 4.18, \(\An[C] \to \lra{C \mid \Sigma}\) is universal among logos morphisms \(\phi\colon \An[C] \to T\) satisfying \(\Sigma^c \subseteq \ker(\phi)\). But since \(\ker(\phi)\) is a congruence, this is equivalent to the condition that \(\Sigma \subseteq \ker(\phi)\), i.e. that \(\phi\) inverts \(\Sigma\).