(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
Pre-additive categories are part of a sequence of additive and abelian categories.
The term ‘pre-additive category’ is often used for any -enriched category, for example in section 8 of Kashiwara–Schapira Categories and Sheaves, in Weibel’s An introduction to homological algebra and Popescu’s Abelian categories with applications to rings and modules.