David Lewis's counterpart theory and Haecceitism debates show that world-level isomorphism and individual-level property preservation can come apart depending on one's theory of trans-world identity.

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Counterpart theory(as used in metaphysics and modal logic): A way of thinking about possible worlds where instead of asking 'could I have been different?', you ask 'is there a version of me in another possible world that is different?' That other version is your 'counterpart.'
David Lewis(the philosopher who created this theory): An influential American philosopher (1941-2001) who developed Counterpart Theory as a way to understand how we talk about objects in different possible worlds.
Haecceitism: A theory of primitive identity of individuals across possible worlds.
Individual-level property preservation(A way of asking whether the same individual retains their defining features): Whether a specific person or thing keeps the same qualities or characteristics when we imagine different scenarios or possible worlds.
Trans-world identity(The core debate: how do we know if it's really the same individual in a different possibility?): The question of what makes someone or something the same person or thing when we compare them across different possible worlds or scenarios.
World-level isomorphism(A way of comparing whether two different possible worlds are the same in structure): Two possible worlds being structurally identical—meaning they have all the same events, objects, and relationships arranged in the same way.

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