Proposition 4.30.
The category \(\Logos\) admits small limits, and the inclusion \(\Logos\hookrightarrow\Cat\) preserves them. Equivalently, \(\Topos\) admits small colimits.
Proof
Given a diagram \(I \to \Logos, i \mapsto T_i\) of logoi, define \(T := \lim_i T_i\), which is again a presentable category. Given an indexing diagram \(K\), a functor \(K^{\triangleright} \to T\) is a colimit diagram if and only if it becomes a colimit diagram in each \(T_i\). The analogous statement holds for finite limit diagrams. In particular, colimits and finite limits in \(T\) are computed pointwise in the categories \(T_i\). Note that we have
\[\Ar(T) \simeq \lim_i \Ar(T_i) \qquadtext{ and } \Ar^{\pb}(T) \simeq \lim_i \Ar^{\pb}(T_i).\]
It then follows immediately from the characterization of logoi given in Proposition 2.16 that \(T\) is again a logos. Furthermore, we see that a functor \(S \to T\) is a logos morphism if and only if each composite \(S \to T_i\) is a logos morphism. It follows that the limit in \(\Cat\) is in fact a limit in \(\Logos\), as desired.