A relational variable set on a category \({\mathcal B}\) is a lax functor \({\mathcal B}\to{\mathfrak{Rel}}\), where \({\mathfrak{Rel}}\) is the partially-ordered 2-category of sets and relations. It is shown that there is a 2-adjunction between \({\mathfrak{Rel}}^{\mathcal B}\) and \({\mathfrak{Cat}}/{\mathcal B}\), which yields an equivalence the locally preordered 2-category \({\mathfrak{Rel}}^{\mathcal B}\) and the slice 2-category \({\mathfrak{Cat}}_f/{\mathcal B}\) of faithful functors over \({\mathcal B}\). It is also shown that a faithful functor \(p: {\mathcal E}\to {\mathcal B}\) is exponentiable in \({\mathfrak{Cat}}_f/{\mathcal B}\) if and only if the corresponding relational variable set \({\mathcal B}\to{\mathfrak {Rel}}\) is non-unitary functor, or equivalently a certain weak factorization lifting property (WFLP) holds. An application to the construction of the simplification of a dynamic set with respect to a change in time domains is discussed.