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We read in enriched category:

A category enriched in Set is a locally small category.

I do not see why. It seems to me that, actually, a category enriched in Set is only a locally small protocategory but not necessarily a category. Indeed, a category is a directed graph in that each morphism has a unique source and a unique target and thus hom-sets are disjoint, but I do not see which axiom of enriched categories in Set constrains hom-sets to be disjoint.

Bruno
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    (a) Yes indeed a Set-enriched category is locally small and nobody claims otherwise, and if not it is just because the word "locally small" is usually omitted, (b) there is no practical difference between categories with disjoint hom-sets and categories without. See the linked duplicates. This means that nothing new is here to be added and I close the question as a duplicate. – Martin Brandenburg Jan 04 '24 at 10:39
  • @MartinBrandenburg (a) has nothing to do with my question. (b) : I am not asking about "practical" differences between protocategories and categories. I am asking if the formal correct statement should be "an enriched category in Set is a category" or "an enriched category in Set is a protocategory" and I did not see the answer in the links that are mentioned. – Bruno Jan 04 '24 at 12:22
  • (a) It has, because you explicitly stated "locally small" as a missing property. (b) It depends on the definition of a category. You are using one where hom-sets are required to be disjoint. There are other ones where this is not the case, and in fact where the definition of a category exactly coincides with the notion of a category enriched over Set. More details in the linked duplicates. By the way, nobody uses the term "protocategory" these days. The nlab article is more of interest for the history of mathematics. – Martin Brandenburg Jan 04 '24 at 12:24
  • @MartinBrandenburg (a) : No, I never wrote that an enriched category in Set is not locally small, I even wrote the opposite : " a category enriched in Set is only a locally small protocategory". Prof. Johnstone, who wrote the Elephant books, uses the word "protocategory". My terminology is consistent : I use, in this question, only the nlab terminology. If you use another definition of "category", then my question becomes irrelevant, but you should not change the terminology in order to try to justify to close the question. – Bruno Jan 04 '24 at 12:39

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