 Algebraizable logics can be characterized by the existence of this kind of isomorphism between congruences and logic filters on the algebras of their algebraic counterpart. To spell out this characterization we need a couple of definitions. Let \(\bL\) be a logic. The Leibniz operator on an algebra \(\bA\) (relative to \(\bL)\) is the map from the \(\bL\)-filters of \(\bA\) to the set of congruences of \(\bA\) that sends every \(\bL\)-filter \(D\) of \(\bA\) to its Leibniz congruence \(\bOmega_{\bA }(D)\). We say that the Leibniz operator of a logic \(\bL\) commutes with the inverses of homomorphisms between algebras in a class \(\bK\) if for every homomorphism \(h\) from an algebra \(\bA \in \bK\) to an algebra \(\bB \in \bK\) and every \(\bL\)-filter \(D\) of \(\bB, h^{-1}[\bOmega_{\bB }(D)] = \bOmega_{\bA }(h^{-1}[D]\)).