It was an English duke who died of exhaustion (the French suffered far more than just two notable fatalities), although I'm sure many of the French died that way too. They were so tightly packed by the flanking longbowmen and their own numbers that they literally had no space to fight properly and trampled each other to death in a rather messy way.
Edit: While military accounts are often exaggerated, there's little reason to doubt that the Spanish were outnumbered and still cut their way through large numbers of poorly-armed Aztec warriors. Horsemen were intimidating enough to musketeers armed with armor-piercing weapons (then again, accounts mention musket balls rattling off the thick cuirasses and helmets of the heavy cavalry at Waterloo, so maybe not THAT armor piercing), they must have been utterly terrifying even to the best Aztec troops.
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