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I have a unit with a character in it. The unit gets charged by a daemon price and fateweaver. Both make it into combat. My opponent elects not to challenge. I issue a challenge with my character. Since they're different units, can I choose to challenge either Fateweaver or the daemon prince, or do I simply issue a challenge, and my opponent gets to decide who accepts?

IIRC you pick the challenger when you declare your challenge, then your opponent picks who accepts if he does accept. If he doesn't you pick one of his characters to not be able to fight.

It is quite clearly laid out on the page 64 though

You do not get to pick who you are challengING, you get to pick your challengER.

ONLY if they refuse would you be able to then pick who is "retired" from the fight.

And in this case neither can refuse, as they are both units of one.

You are not issueing a challenge to a unit or specific character, you are simply issueing a challenge to the enemies in close combat with you.
So your opponent in this case has 2 units of single characters, so he has to accept the challenge with one of them, but he chooses which one.