# Category with endomorphisms only

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