Just want to make sure I’ve understood the proper use of the talent -

For each rank of Adversary, I upgrade the difficulty of a check targeting that enemy once. Difficulty dice are upgraded to Challenge dice first, then if there are unused upgrades, add Difficulty dice to the pool. For example, a mage attacks an enemy with the Adversary 2 talent at short range with an attack spell - the standard difficulty would be one Difficulty die, but with the Adversary 2 talent, the difficulty becomes one Challenge and one Difficulty die, correct?

You are correct. However,

11 minutes ago, tafattel said:

For each rank of Adversary, I upgrade the difficulty of a check targeting that enemy once.

It is specifically limited to combat checks, not any check.

37 minutes ago, tafattel said:

the standard difficulty would be one Difficulty die, but with the Adversary 2 talent, the difficulty becomes one Challenge and one Difficulty die, correct?

Yes, and with three ranks the pool would be two challenge dice, and with four ranks two challenge and one difficulty. If the player can downgrade the check after that, you turn red dice into purple and do nothing with the purple dice.

In your example, if the mage had a talent that would let him downgrade the difficulty of the check twice, he'd only turn the one red die into a purple one, the pool would still have two purple dice in it.