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How is called a category with endomorphisms only?

How is called a subcategory got from an other category by removing all morphisms except of endomorphisms?

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If the only endomorphisms are the identities one calls it the trivial category on the underlying set. – Julian Kuelshammer May 29 '13 at 21:21
@JulianKuelshammer any poset regarded as a category has no not-identity endomorphisms, but the resulting category is not trivial on its underlying set. – Ittay Weiss May 29 '13 at 21:25
@IttayWeiss Of course if the category satisfies both: having only endomorphisms as morphisms, and having only identities as endomorphisms. Another name for that seems to be discrete category. – Julian Kuelshammer May 29 '13 at 21:27

Every category in which every arrow is an endomorphism is a coproduct (in the category of categories) of monoids (a monoid is a category with just one object). So a category with all morphisms endormophisms is a coproduct of monoids. I'm not aware of any specific terminology for it.

Clearly the category of all coproducts of monoids admits an inclusion functor into the category of all categories. The construction you describe (of removing all non-endo morphisms) describes a right adjoint to this inclusion functor.

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