Agatha's Soul Cauldron being ignored by system
Agatha's Soul Cauldron can exile a card from a graveyard at instant speed.
But... When someone else is playing a card that puts cards into their graveyard and then takes them out again, they are getting to do this without passing priority to Agatha's Soul Cauldron. So, she cannot exile the card before they pull it out of their graveyard.
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[Deleted User] commented
It does work that way. But there are also cards that prohibit targeting cards in graveyards, most notably in current Standard is Dennik.
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danius99#53743 commented
Allow Agatha's Soul Cauldron to target cards in your own graveyard, as described in the card's text.
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[Deleted User] commented
If the cards are put into the graveyard and then out of it during the resolution of a spell or ability, then NO player has priority to do anything other than what the spell/ability asks for. After it finishes resolving, the active player will receive priority (after state based actions have been dealt with, and triggers have been put on the stack, of course). You can't interrupt a spell's/ability's resolution.