In a commutative diagram of two exact five-term sequences, if the first and fourth vertical maps are epimorphisms and the second and fifth are monomorphisms, then an isomorphism in the middle follows; in particular, four surrounding isomorphisms force the fifth.
For commuting five-term exact sequences in an abelian category, the first vertical map being epic and the second invertible give an invertible induced map on the images entering the middle objects, because these images are the cokernels of the first horizontal maps. The fourth vertical map being invertible and the fifth monic similarly give an invertible map on the images leaving the middle objects, because these are the kernels of the last horizontal maps. The image factorization in an abelian category packages each middle object between these two images in a short exact sequence. The short five lemma then makes the middle vertical map invertible. The proof is categorical and does not require an embedding into a category of modules.
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The Five Lemma is a result in the field of homological algebra, particularly in the context of derived categories and spectral sequences. It provides a criterion for when a five-term exact sequence of chain complexes splits. This lemma is commonly used in the study of abelian categories and the derived functor theory.