With regard to the confusion over terminology, the term "enriched category" is an example of what some people know as the [Mathematical Red Herring Principle](https://ncatlab.org/nlab/show/red+herring+principle):

> **Mathematical Red Herring Principle** A “red herring” need not, in general, be either red or a herring.

Just as John said these things are not generally categories, so too there is not necessarily anything you might recognise as enriching going on. Well, maybe it would be fair to think of these as enriched sets :-)

The terminology came about when the early examples of enriched categories that people considered were actually categories which had been 'enriched', but the terminology never changed when people realised that something much more general was going on.

The confusion exists amongst high-level mathematicians as well: I have been to a couple of talks recently in which a geometer or topologist has said "A \\(\mathcal{V}\\)-category is a category such that..." So don't feel bad if you think that the terminology is daft!

> **Mathematical Red Herring Principle** A “red herring” need not, in general, be either red or a herring.

Just as John said these things are not generally categories, so too there is not necessarily anything you might recognise as enriching going on. Well, maybe it would be fair to think of these as enriched sets :-)

The terminology came about when the early examples of enriched categories that people considered were actually categories which had been 'enriched', but the terminology never changed when people realised that something much more general was going on.

The confusion exists amongst high-level mathematicians as well: I have been to a couple of talks recently in which a geometer or topologist has said "A \\(\mathcal{V}\\)-category is a category such that..." So don't feel bad if you think that the terminology is daft!