Fun and Winter
Fun and winter is what defines Jack Frost. He is the guardian of fun and he uses his winter powers to bring fun to the children, not so much adult.
Winter is one of the four seasons of the year and it’s the coldest one. It starts with the winter solstice and then ends on the vernal equinox (normally from 21st Dec to 21st March - Now that’s for the northern hemisphere). And Jack Frost is one of the personifications of the winter, or at least that’s what Wikipedia says. Anyway, with different cultures the start and how long it lasts is different, it’s also different how they perceive it.
Definition of fun: what provides amusement or enjoyment; specifically: playful often boisterous action or speech <full of fun>
This definition is what Jack Frost is, he is playful and loves the amusement,
and loves to share it with the others.