I just found out that Marius Furter has a whole series of lectures on enriched category theory! This is really fantastic.

This looks wonderful. There is a real shortage of good introductory material on enriched category theory.

To me the subject of enriched categories jumps from being â€śtrivialâ€ť (perhaps mindblowing, but really a straightforward generalization of ordinary category theory) to â€śnontrivialâ€ť when it reaches the subject of limits and colimits. Because thereâ€™s usually not a â€śdiagonalâ€ť enriched functor \Delta: C \to C^D where C is the enriched category weâ€™re trying to take limits or colimits in and D is the â€śdiagramâ€ť enriched category, the usual trick of defining limits and colimits as right and left adjoints to \Delta: C \to C^D doesnâ€™t work, nor even the whole idea of a cone or cocone. So we must turn to weighted limits and weighted colimits. So it would be really nice to have a gentle, expository introduction to enriched category theory that covers these.