The notion of a continuous $MV$-valued L-fuzzy relation in Chang topological L-fuzzy spaces is defined, and the category $\bf RTop$ of these spaces with continuous L-fuzzy relations as morphisms is presented. It is proven that in category $\bf RTop$ there exist fuzzy products, which are modifications of the classical products in categories. Two special subcategories of $\bf RTop$ are presented, defined using the category of L-fuzzy approximation spaces and the category of L-fuzzy partitions, both with L-fuzzy relations as morphisms. For these two subcategories, we show how the concept of a continuous algebraic L-fuzzy relation (e.g., the concept of an approximate sum) can be defined on the objects of these categories using the fuzzy products in these subcategories.