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Dun: Mayflies have two adult stages. They first emerge from the water as duns (scientifically known as the subimago stage). They then molt into the spinner (imago) stage, in which they mate and die. Sometimes the word "dun" is confusingly used to refer to a brownish gray color in fly tying materials.
Molting: When aquatic insects with hard exoskeletons (like mayfly and stonefly nymphs) grow bigger, their exoskeleton does not grow with them. Instead they grow a new, larger one underneath and shed the old one when it's too small. This process is called molting.
Spinner: There are two winged stages of adult mayflies. They emerge from the water as duns, molt on land (usually) into their fully mature stage, spinners. As spinners, they mate, lay eggs, and die.