Left exact localizations and congruences

The equivalence \((1) \iff (3)\) follows immediately from Definition 4.14 and Lemma 4.15, combined with Proposition A.15.The implication \((3) \implies (4)\) follows since \(K\) is the kernel of \(L\colon T \to T[K^{-1}]\), see Proposition A.15. For \((4) \implies (2)\), if \(K\) is the kernel of a morphism of logoi \(\phi\), then \(K\) is closed under finite limits because \(\phi\) is left exact. Furthermore, since \(\phi\) is a morphism of logoi, it preserves colimits, and thus its kernel is strongly saturated of small generation by [Lurie 2009, Proposition 5.5.4.16] (see Proposition A.14).Finally, for \((2) \implies (1)\), it suffices to note that a class closed under finite limits is in particular closed under base change.