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Hexbear: A grizzly, with literally 50% more bear. Trundles around on all sixes most of the time, doing bear stuff, but has the flexibility to rear back into a functional bear-taur. Also 50% better at doing bear-related activities (including slapping your face into the next clearing), but lower birth rate at only one cub per season.
Hihi: Ape-like creatures with stone masks for faces, found exclusively in the wildest forests of the Yamato region. Guardians of the forest, they have the strength of ogres, and the mysterious ability to slide trees around the landscape as if they were on casters. Their preferred M.O. is to trick people into leaving their forests expeditiously, but will fight as needed.
Hippogryph: the trendy exotic mount of the highborn. Has the front half of a bird of prey, wings, and the back half of a horse. Hideously expensive to raise, bad tempered, finicky, have an irrational taste for horseflesh, but they do fly.
Horse: Big ungulates with shoulders, making them excellent for agriculture. Also large enough and fast enough to be ridden in combat. Most Earth breeds can be found someplace.
Humminggryphon: A gryphon the size of a sparrow, with the front of a hummingbird, and the back of a black footed cat. Largely harmless, they are trainable, respond well to familiar-bonding magics, popular as pets among the highborn, and used as messengers in the Hazad Caliphate. Humminggryphons have a much higher flight endurance than one would expect.
Huorn: One of those great, spiritually revered trees that has gotten corrupted by some means, and has decided to lash out at everyone around it. Assuming “it's only a tree” will get you quite dead: a Huorn has a lot of spiritual power backing it up.
Hydra: Any drake with more than one head. The ones that grow back heads and necks when you chop them off are rare, but not unheard of.