Convergence theory studies relations between filters and points. Continuity assures existence of initial and final convergences. This framework enables us to define functors objectwise, like in topological constructs, but a slightly higher level of abstraction in the latter case makes the formalism much more complex. The extension of the concept of adherence to arbitrary families makes it possible to treat various reflective classes of convergence as special cases of types of compactness of families, and not only of sets. This approach enables one to see various classes of quotientness and perfection of maps as forms of compactness of corresponding relations. Our approach has the advantage to represent classical topological properties as solutions to functorial inequalities. Is there any point to consider relations between arbitrary isotone families, not only filters, and points? Greco’s theory of limitoids constitutes the affirmative answer to this question. A limitoid is a functional $T:L^X \rightarrow L$, where $L$ is a complete lattice and $X$ is a set, which is isotone, and commutes with lattice complete homomorphisms. If $L$ is completely distributive, then each limitoid can be represented as a lower limit along an isotone family, which, in general, is not a filter. As all the variational limits of De Giorgi are limitoids, Greco’s theory is a powerful tool for the latter. But as contours are, in fact, lower limits over families of sets, limitoids apply to diagonality and to regularity in convergence theory.