Local homology groups were first introduced by E. R. van Kampen thirty years ago. Since then, numerous variants and generalizations of van Kampen's definition have been given by various authors。In a recent paper of To R. Brahane, an axiomatic approach to the local homology theory is given and the uniqueness theorem is established for the category of triangulable local pairs. His methods of proof are parallel to these used by Eilenberg and Steenrod in the global homology theory. If one applies the global uniqueness theorem of Eilenberg and Steenrod instead of paraphrasing their methods, the proof of the local uniqueness theorem would be much simpler. In the present technical note, such a proof is produced.
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