A characterization of left exact localizations

The implications (1) \(\implies\) (2) \(\implies\) (3) are clear. For \((3) \implies (1)\), let \(K\) be the class of morphisms inverted by \(L\), and assume \(K\) is closed under base change. Observe that an object of \(C\) lies in the essential image of \(R\) if and only if it is local with respect to \(K\). Since the terminal object \(1\) of \(C\) is clearly \(K\)-local, it follows that \(L(1) \iso 1\), and so \(L\) preserves the terminal object. It remains to show that \(L\) also preserves pullbacks, so consider two morphisms \(X \to Y \leftarrow Z\). Since \(K\)-local objects are closed under limits in \(C\), it follows that \(RLX \times_{RLY} RLZ\) is \(K\)-local; in particular, it defines the pullback in \(D\). It now remains to show that the canonical map \(X \times_Y Z \to RLX \times_{RLY} RLZ\) lies in \(K\). We may factor this map as a composite The second and third maps are base changes of morphisms in \(K\), hence lie in \(K\) again by assumption. The first map is a base change of the diagonal of \(Y \to RLY\), which is therefore a section of the projection \(\pr_1\colon Y \times_{RLY} Y \to Y\). Since \(K\) is closed under 2-out-of-3, it suffices to show that \(\pr_1\) is in \(K\), which is true as it is a base change of \(Y \to RLY\).