Notation 5.63.
Let \(\varphi\colon T \to S\) be a morphism of logoi with kernel \(K := \ker(\varphi)\). Then we have an inclusion \(K^{\mono} \subseteq K\), giving rise to three morphisms of logoi as follows:
\[T \xrightarrow{\,\varphi^{\mono}\,} T/K^{\mathrm{mono}} \xrightarrow{\,\varphi^{\epi}\,} T/K \xrightarrow{\varphi^{\cons}} S.\]
We refer to this as the quotient triple factorization of \(\varphi\). We further write
\[\varphi^{\quottext} := \varphi^{\epi} \circ \varphi^{\mono} \qquadtext{ and } \varphi^{\wcons} := \varphi^{\cons} \circ \varphi^{\epi}.\]