 Theorem 4.  A logic \(\bL\) is algebraizable if and only if for every algebra \(\bA \in \bAlg\bL\) the Leibniz operator commutes with the inverses of homomorphisms between algebras in \(\bAlg\bL\) and is an isomorphism between the set of all \(\bL\)-filters of \(\bA\), ordered by inclusion, and the set of congruences \(\theta\) of \(\bA\) such that \(\bA/\theta \in \bAlg\bL\), ordered also by inclusion.