nLab reflective product-preserving sub-(∞,1)-category - internal formulation

Idea

The external definition of reflective sub-(∞,1)-category via the universal property of the reflector has an immediate formulation in the internal language of an (∞,1)-topos. This internal formulation, however, automatically gives a reflective product-preserving sub-(∞,1)-category.

References

HoTT-Coq code for internal reflective subcategories is at

• HoTT/Coq/ReflectiveSubcategory.v

The fact that in the internal formulation reflective subcategories are automatically product-preserving is mentioned on p. 5 of

• Eugenio Moggi, A note on SDT in Filter spaces (pdf)