On 5 Dec 2019, at 7:38 AM, Paul Taylor <cats@PaulTaylor.EU<mailto:cats@paultaylor.eu>> wrote: - I would like the homs of C to be objects of E, so it is an E-enriched category. Dear Paul If, rather than C E-enriched, you give meaning to E-indexed families of C objects by taking C to be a category (parametrized) over E, then what I think you want is in the works of B\'enabou, Par\'e-Schumacher [SLNM 661], and Section 2 of 32. (with D. Schumacher) Some parametrized categorical concepts, Communications in Algebra 16(1988) 2313--2347 where you will find precise references to other works. For example, let C just be the topos E. We know how to internalise ("classify") subobjects - using the subobject classifier Omega. Taking C to be the the arrow category of E, over E via the codomain functor, wellpowered is about the existence of a subobject classifier in E. All the best, Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]