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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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    I very much doubt there is a name for such a concept.2012-06-14

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Why should such a morphism have a special name? This property is a property of the domain, it does not depend on the specific morphism. Maybe you are looking for emptiness of an object, it is quite similar, but this notion requires the object to be an initial object (that would explain why there are that many of those morphisms ;)) and allows some additional isomorphisms.

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Maybe you should work out the situation in categorical terms, so we can see what is going on. To me it just means, that in your category $\mathbf{C}$ there are objects $B$, $C$ (with $B \neq C$ necessarily) with $hom(B,C) \neq \emptyset$ such that $hom(A,B) = \emptyset$ for all objects $A \neq B$. Is that what you mean?

In that case maybe you should look into graph theory. If you look on the category as a graph this means, that the vertex $B$ has no predecessors And maybe this has a name.

However from a category theoretic point of view this is nothing special and hence I don't think there is a special name for it.

Looking on your comments I think you would do well to study category theory more deeply. A category 'without precomposition' doesn't make sense at all. If you arrive at situations like that, it is usually an indicator that you should rethink what you are doing.

Moreover if you put constrains on SOME morphisms from $hom(B,C)$ you leave category theory and what is called 'natural' ...

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    From the (directed) graph theoretic perspective, the name "source" might be appropriate. A source in a directed graph is a vertex that is not the head (=target) of any edge.2012-06-14