We develop a theory of categories which are simultaneously (1) indexed over a base category $S$ with finite products, and (2) enriched over an $S$-indexed monoidal category $V$. This includes classical enriched categories, indexed and fibered categories, and internal categories as special cases. We then describe the appropriate notion of ``limit'' for such enriched indexed categories, and show that they admit ``free cocompletions'' constructed as usual with a Yoneda embedding.
Keywords: monoidal category, enriched category, indexed category, fibered category
2010 MSC: 18D20,18D30
Theory and Applications of Categories, Vol. 28, 2013, No. 21, pp 616-695.
http://www.tac.mta.ca/tac/volumes/28/21/28-21.pdf
Revised 2022-03-28. Original version at
http://www.tac.mta.ca/tac/volumes/28/21/28-21a.pdf