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I recently stumbled upon the book Derived Functors in Functional Analysis by Wengenroth. In it, he defines semi-abelian categories quite differently from the nlab:

An $\mathsf{Ab}$-category $\mathsf A$ is called semi-abelian if it admits binary products and every arrow $f$ has a kernel and a cokernel, and the induced arrow $\mathrm{Coim}\rightarrow \mathrm{Im}f$ is a bimorphism.

Is there a standard term, or systematic study of such categories?

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