Quantaloid

In mathematics, a quantaloid is a category enriched over the category Sup of "suplattices". In other words for any objects "a", "b" the morphism object between them is not just a set but a complete lattice, in such a way that composition of morphisms preserves all joins:
formula_1
The endomorphism lattice formula_2 of any object formula_3 in a quantaloid is a quantale, whence the name.