Corollary 2
Tag 0DRL, permanent. Cite it as [Bourbaki, Tag 0DRL].
Algebra, chapter X, § 2. Complexes de A-modules
Let u be a morphism of complexes. If Ker u and Coker u have zero homology, then u is a homologism.
Indeed, let u : E → E’ be a morphism of complexes. If Ker u (resp. Coker u) has zero homology, then the canonical morphism E → Im u (resp. Im u → E’) is a homologism by Corollary 1.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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