= Five lemma via image factorization
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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