  Morphisms between (formal) contexts are certain pairs of maps, one between
objects and one between attributes of the contexts in question. We study
several classes of such morphisms and the connections between them. Among other
things, we show that the category CLc 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. On the other hand, we obtain
a dual adjunction between CLc and the category of contexts with so-called
concept continuous morphisms. Suitable restrictions of the adjoint functors
yield a categorical equivalence and a duality between purified contexts and
doubly based lattices, and in particular, between reduced contexts and
irreducibly bigenerated complete lattices. A central role is played by
continuous maps between closure spaces and by adjoint maps between complete
lattices.
