Characterizations of congruences

The equivalence between (1) and (2) is the definition. A congruence is closed under finite limits in \(\Ar(T)\), see [Anel et al. 2022, Proposition 4.2.3], which gives (2)\(\Rightarrow\)(3). Clearly (3) implies (4). Conversely, finite-limit closure implies stability under base change, since a base-change square is a pullback in \(\Ar(T)\). Moreover, Lemma 5.34 gives left cancellation. Saturation already gives composition and right cancellation, so \(K\) satisfies 2-out-of-3. This proves (4)\(\Rightarrow\)(2).Condition (2) implies (5): the projection \(X\times_YX\to X\) is a base change of \(f\), and its section \(\Delta_f\) belongs to \(K\) by 2-out-of-3. Conversely, assume (5). Since \(K\) is saturated, it is closed under composition and has right cancellation. It remains to check left cancellation. Given \(X \xrightarrow{f} Y \xrightarrow{g} Z\) with \(g, gf \in K\), factor \(f\) as The second map is a base change of \(gf \in K\), hence in \(K\). The first is a base change of \(\Delta_g\), which is in \(K\) by assumption. Thus \(f \in K\).