  The present work presents some results about the categorial relation between
logics and its categories of structures. A (propositional, finitary) logic is a
pair given by a signature and Tarskian consequence relation on its formula
algebra. The logics are the objects in our categories of logics; the morphisms
are certain signature morphisms that are translations between logics
(\cite{AFLM1},\cite{AFLM2},\cite{AFLM3} \cite{FC}). Morphisms between
algebraizable logics (\cite{BP}) are translations that preserves algebraizing
pairs (\cite{MaMe}): they can be completely encoded by certain functors defined
on the quasi-variety canonically associated to the algebraizable logics. This
kind of results will be useful in the development of a categorial approach to
the representation theory of general logics (\cite{MaPi1}, \cite{MaPi2},
\cite{AJMP}).
