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 the pair of maps , is (jointly) strongly epimorphic.
This terminology is introduced in
gorical Structures 4 (1996) 302-327
Exposition is in the section 1.2 in
Unfortunately the terminology is not compatible with the notions of unitality of A-infinity categories.
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