nLab
unital category

Definition

A category is unital in the sense of Bourn if it has a zero object (is a pointed category), admits finite limits and for all objects $X,Y$ the pair of maps $({\mathrm{id}}_X,0):X\to X\times Y$, $(0,{\mathrm{id}}_Y):Y\to X\times Y$ is (jointly) strongly epimorphic.

References

This terminology is introduced in

• Dominique Bourn, Mal’cev categories and fibrations of pointed objects, Appl. Cate- gorical Structures 4 (1996) 302-327

Exposition is in the section 1.2 in

• Francis Borceux, Dominique Bourn, Mal’cev, protomodular, homological and semi-abelian categories , Mathematics and Its Applications 566, Kluwer 2004

Unfortunately the terminology is not compatible with the notions of unitality of A-infinity categories.