Colimits of logoi exist

Let \(T_{\bullet}\colon I\to\Logos\) be a small diagram. By uniformization for accessible categories and functors, we may choose a regular cardinal \(\kappa\) such that each \(T_i\) is \(\kappa\)-accessible, every transition morphism is strongly \(\kappa\)-accessible, and the full subcategory of \(\kappa\)-compact objects is closed under finite limits. The transition morphisms therefore restrict to a diagram \(C_{\bullet}\colon I\to\Cat\) of small categories with finite limits. For the simultaneous choice of \(\kappa\), see the uniformization argument in [Lurie 2009, Remark 5.4.2.13]; closure of the compact objects under finite limits follows after a further enlargement from [Lurie 2009, Proposition 5.4.7.4].For each \(i\), the colimit extension \(\PSh(C_i)\to T_i\) of the inclusion \(C_i\hookrightarrow T_i\) is left exact by Proposition 2.43. Its right adjoint is the restricted Yoneda functor, which is fully faithful because \(C_i\) is a collection of generators. Composing this localization with the canonical localization \(\An[C_i]\to\PSh(C_i)\) from the proof of Proposition 4.26, we obtain a left exact Bousfield localization Choose a small class of morphisms \(\Sigma_i\subseteq\An[C_i]\) whose generated congruence is \(\ker(q_i)\). Thus \(q_i\) induces an equivalence \(\lra{C_i\mid\Sigma_i}\simeq T_i\).These presentations are compatible with the diagram. Indeed, for an arrow \(a\colon i\to j\) in \(I\), let \(a^*\colon T_i\to T_j\) denote the transition morphism and also its restriction \(C_i\to C_j\). The two logos morphisms given by \(a^*q_i\) and by \(q_j\An[a^*]\) agree on the generators \(C_i\), hence are equivalent by the universal property of the free logos. It follows that \(\An[a^*]\) sends \(\ker(q_i)\) into \(\ker(q_j)\) and therefore induces the original transition morphism between the presented logoi.Now form the small category \(C:=\colim_iC_i\), and let \(\Sigma\) be the union of the images of the small classes \(\Sigma_i\) in \(\An[C]\). Consider the logos The functors \(C_i\to C\) induce a cocone \(T_i\to T\). To verify its universal property, let \(S\) be a logos. Then Under this equivalence, a logos morphism \(\An[C]\to S\) inverts \(\Sigma\) if and only if each of its restrictions to \(\An[C_i]\) inverts \(\Sigma_i\). By Lemma 4.25, such data are precisely a compatible family of logos morphisms \(T_i\to S\). Hence so \(T\) is the colimit of \(T_{\bullet}\) in \(\Logos\).