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We prove that the category of F-coalgebras admits exact sequences, under condition F preserves pullbacks; the underlying category 풞 is assumed to be cocomplete with epi-(regular mono) factorizations, finite products and exact sequences. [ABSTRACT FROM AUTHOR]

It is well known that the classical diagram lemmas of homological algebra for abelian groups can be generalized to non-abelian group-like structures, such as groups, rings, algebras, loops, etc. In this paper, we establish such a generalization of the "salamander lemma" due to G. M. Bergman, in a self-dual axiomatic context (developed originally by Z. Janelidze), which applies to all usual non-abelian group-like structures and also covers axiomatic contexts such as semi-abelian categories in the sense of G. Janelidze, L. Márki and W. Tholen and exact categories in the sense of M. Grandis. [ABSTRACT FROM AUTHOR]