How a quasi-uniformity on an abstract category — a family of endomaps on each object's lattice of subobjects, equivalently a co-perfect syntopogenous structure, equivalently a family of closure operators — can be transported along a functor. You learn to define continuity of a morphism, and then of a functor, with respect to two such structures, and to construct the coarsest structure making a pointed endofunctor's unit, an M-fibration, or a left adjoint continuous, with the finest structure arising dually from a copointed endofunctor. The constructions recover the classical pullback closure operator and, in concrete examples, the initial quasi-uniformities on T0 quasi-uniform spaces, uniform spaces and Hausdorff topological groups.
A quasi-uniformity on a set is classically a family of entourages; this paper works with its categorical form. Fix a category C with a proper (E,M)-factorization system, M-complete, so each object X …