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' ...