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New entry homological category.
Probably in the first line you meant to write “homological” instead of “regular”.
Of course it is not. It is not an enriched category, but an ordinary one category satisfying some conditions. Of course, the homological algebra is then done with an appropriate notion of complexes in it and the five lemma is a lemma in the original category. Typical examples are categories of algebras over certain subclass of algebraic theories in Set. Every semi-abelian category is an example. There is also a generalization, relative homological categories.
Jim, I and mislead and do not like the terminology in this case either. It is misleading. Bourn and Janelidze thank Johnstone for the help with terminology, and my experience in past is that Johnstone often has strange inventions in terminology.
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