Definition 4.24.
A logos presentation is a pair \((C,\Sigma)\) consisting of a small category \(C\) together with a small class of morphisms \(\Sigma\) in \(\An[C]\). We define the logos presented by \((C,\Sigma)\) as the quotient
\[\lra{C \mid \Sigma} \quad := \quad \An[C] \, / \, \Sigma^c\]
of the free logos on \(C\) by congruence \(\Sigma^c\) from Example 4.22. In particular, \(\lra{C \mid \Sigma}\) is a logos and the quotient map \(\An[C] \to \lra{C \mid \Sigma}\) is a logos morphism.
We wish to think of \(C\) as the `generators' of \(\lra{C \mid \Sigma}\), and of \(\Sigma\) as the `relations'.
More explicitly, \(\lra{C \mid \Sigma}\) may be identified with the full subcategory of \(\PSh(C^{\lex})\) spanned by the \(\Sigma^c\)-local presheaves.