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Once we have fixed an isomorphism A — >X , it is harmless to treat A and X as the same object, precisely because we have the maps f and f-1 to 'translate."
--- Lawvere, Schnauel. 2009.
It's extremely useful in most cases, but can cause mental errors when an isomorphic association creates an unstated assumption of unity, i.e. "the same object." A classic example would be the isomorphism between glory and mortal combat as described by Homer in the Iliad.