The abstract theory of algebraizable logics is a very poweful tool for studying a given target logic using techniques of universal algebra. However, this theory has some applicability limitations. One important example is the class of non-truth-functional logics, in particular the paraconsistent systems of daCosta. In this talk, we overview the main concepts and results of the theory of algebraizable logics and present an alternative characterization of these notions in terms of maps between the target logic and unsorted equational logic. Using this characterization, we then propose an extension of the theory where the role played by unsorted equational logic can be replaced by any other base logic that satisfies a few reasonable requirements. We illustrate the approach by exploring the non-truth-functional paraconsistent system C1 of daCosta using two-sorted equational logic as a base.