Correspondences of the Semigroup of Endomorphisms of an Equivalence Relation

In terms of different constructions of the wreath product of a transformation semigroup with a small category and of the direct product of wreath products of groups, faithful representations of three correspondences of the semigroup of all endomorphisms of an arbitrary equivalence relation are described, namely, of the semigroup of all endotopisms, of the monoid of all strong endotopisms, and, correspondingly, of the group of all autotopisms of a given equivalence relation.