Productivity of coreflective subcategories of uniform spaces

Using the fact that each product of uniform quotient mappings is a quotient mapping, new conditions are given for the finite and countable productivity of a coreflective sub-class of uniform spaces. Three basis examples of productive coreflective sub-classes are constructed (connected with products of discrete spaces, proximally fine spaces, and uniformly sequentially continuous mappings) and the coreflective hull of metric spaces is shown to be productive if and only if there exists no uniformly sequential cardinal number.