Towards a good notion of categories of logics

We consider (finitary, propositional) logics through the original use of Category Theory: the study of the “ sociology of mathematical objects”, aligning us with a recent, and growing, trend of study logics through its relations with other logics (e.g. process of combinations of logics as fibring [Gab] and possible translation semantics [Car]). So will be objects of study the classes of logics, i.e. categories whose objects are logical systems (i.e., a signature with a Tarskian consequence relation) and the morphisms are related to (some concept of) translations between these systems. The present work provides the first steps of a project of considering categories of logical systems satisfying simultaneously certain natural requirements: it seems that in the literature ([AFLM1], [AFLM2], [AFLM3], [BC], [BCC1], [BCC2], [CG], [FC]) this is achieved only partially.