The goto place for a mathematically formal account would be the book I referred to before, ISBN 9783540627715. Though rigorous is perhaps shallow. A theorem there, that is shown in the slides I referred to above says: The concept lattice of a relation is complete, and any complete lattice arises as the concept lattice of its associated order.

More interestingly in the abstract of arxiv:1407.0512:

> ...We show that the category of complete lattices with complete homomorphisms is (up to a natural isomorphism) a full reflective subcategory of the category of contexts with so-called conceptual morphisms; the reflector associates with each context its concept lattice.

More interestingly in the abstract of arxiv:1407.0512:

> ...We show that the category of complete lattices with complete homomorphisms is (up to a natural isomorphism) a full reflective subcategory of the category of contexts with so-called conceptual morphisms; the reflector associates with each context its concept lattice.