Functorial Properties of the Reticulation of a Universal Algebra

The reticulation of an algebra A is a bounded distributive lattice whose
prime spectrum of ideals (or filters), endowed with the Stone topology, is homeomorphic
to the prime spectrum of congruences of A, with its own Stone topology.
The reticulation allows algebraic and topological properties to be transferred
between the algebra A and this bounded distributive lattice, a transfer which
is facilitated if we can define a reticulation functor from a variety containing A
to the variety of (bounded) distributive lattices. In this paper, we continue the
study of the reticulation of a universal algebra initiated in [27], where we have
used the notion of prime congruence introduced through the term condition
commutator, for the purpose of creating a common setting for the study of the
reticulation, applicable both to classical algebraic structures and to the algebras
of logics. We characterize morphisms which admit an image through the