Schur's lemma for exact categories implies abelian

Abstract

We show that for a given exact category, there exists a bijection between semibricks (pairwise Hom-orthogonal set of bricks) and length wide subcategories (exact extension-closed length abelian subcategories). In particular, we show that a length exact category is abelian if and only if simple objects form a semibrick, that is, the Schur's lemma holds.

Semibricks in extriangulated categories

“…Recently, Enomoto [6] generalizes the notion of simple objects in an abelian category to an exact category and then generalizes Ringel's bijection to exact categories. The notion of a triangulated category was introduced by Grothendieck and later by Verdier [16].…”