<p>In group theory, Fitting-formations are important classes and object of large study. For inverse semigroups, i-formations and i-Fitting classes have been considered, extending the group case. They are related to correspondences of idempotent separating congruences (and of languages contained in the centralisers). The aim of this paper is to characterise the correspondences of congruences (and of languages) associated to i-Fitting-formations of inverse semigroups. Although the known definitions of i-formation and i-Fitting class of congruences (of languages) are natural and appear in some way as dual of each other, they give rise to classes that intersect trivially, thence that are not associated to i-Fitting-formations of inverse semigroups. This led to the search for different correspondences, denoted by i-Fitting systems. The intersection of the class of all i-Fitting systems with the class of all i-formations of congruences (of languages) is then in bijection with the class of all i-Fitting-formations of inverse semigroups. In the case of Clifford semigroups (in particular, of groups), a dual path can be taken and another kind of correspondence of congruences (of languages) is obtained, which is also in bijecton with Fitting-formations of Clifford semigroups (groups).</p>