Is there a term to describe a morphism in a category that cannot be written as a composition of two (non-identity) morphisms? Naively Googling around I haven't found anyone talking about this, so I don't know if there is an established word for this. I really hope it's the word indecomposable, since it describes an morphism that is not a composite. But the adjectives prime, primitive, and irreducible could also work. I mostly want to make sure I'm don't define such morphisms opposing an established precedent.
Furthermore, is there a name for a category where every morphism can be written as a composite of these "indecomposable" morphisms? Something like a Krull-Schmidt category but for the morphisms instead of the objects. This question is motivated by looking at the functor that takes a quiver and gives you the free category on that quiver. I want the category-theoretic term that describes the morphisms in that free category that came precisely from the arrows of the quiver.