From this Math.SE post:
It would be helpful to have a standard term XXX for “a category without the requirement of Hom-sets to be disjoint” and “category got from XXX by adding source and destination object to every morphism”.
This would greatly help to simplify at least 50% of routine definitions of particular categories. Why should we specify a triple every time we define something categorical? It is too much hand-writing.
I wonder when there are no agreement between mathematicians on this terminology.
Hey, readers, can we make up a term and lobby to make it standard terminology?