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 $latex (f;A;B)$ 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?