Concept lattices of $Q$-relations via diagonal and back-diagonal categories

Let $Q$ be a commutative unital quantale and $\text{QRel}$ the ordered category of sets and $Q$-relations. We develop a categorical foundation for formal and property-oriented concept lattices of $Q$-relations via the diagonal category $\text{DQRel}$ and back-diagonal category $\text{BQRel}$. Our main results are: (1) the formal concept lattice construction is induced by a representable functor on $\text{BQRel}$; (2) the property-oriented concept lattice construction corresponds to a representable functor on $\text{DQRel}$; and (3) for regular $Q$-relations, the property-oriented concept lattice of their pointwise tensor product is naturally isomorphic to the tensor product of the respective property-oriented concept lattices.