Example 4.23.
Let \(\phi\colon S \to T\) be a morphism of logoi. Given a congruence of small generation \(K\) in \(T\), we define its preimage as
\[\phi^{-1}(K) \quad := \quad \{g\in \Ar(S) \mid \phi(g) \in K\}.\]
This is again a congruence of small generation. Indeed, consider the composite morphism of logoi
\[S \xrightarrow{\phi} T \xrightarrow{L} T/K.\]
As the kernel of \(L\) is \(K\), the kernel of this composite is precisely \(\phi^{-1}(K)\). By Corollary 4.16, the kernel of any morphism of logoi is a congruence of small generation, which proves the claim.