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A morphism $m$ of a category has the following property:

No morphism (except of the identity morphism) of the category has codomain equal to the domain of $m$. In other words, $m$ cannot be composed on the right.

Are there any terminology about this case?

In fact, I have a category with many such special morphisms.

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    That is impossible, as any object of the category has the identity morphism to itself.2012-06-12
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    (Your second sentence should start with «Np morphism...»)2012-06-12
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    @ZevChonoles: You are right. I need to decide what to do with identity morphisms.2012-06-12
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    @ZevChonoles: I now think that I would do well not to require the non-existence of the morphisms with such codomains, just never compose $m$ on the right (not because this is impossible but I just don't want to compose it on the right and don't want to define what such composition would produce).2012-06-12
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    @ZevChonoles: But this way, I need to prove existence of a category having defined only left composition of $m$ and right composition to be left arbitrary.2012-06-12
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    @ZevChonoles: I've added "except of the identity morphism".2012-06-12
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    I very much doubt there is a name for such a concept.2012-06-14

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