Ethereum 2.0 is a Proof of Proof (PoS) beacon chain that has its supply coming fro Ethereum 1.0 through a storage contract which will destroy the ETH in the original blockchain and generate the same number of ETH in the beacon chain. This will not be a hardfork as there won't be two ETH in the same time, but rather a migration of ETH from one chain to the other.
In the beginning of Ethereum 2.0 the ETH will not be fully migrated, but will be rather locked and pledged to the new beacon chain. In time probably this will be settled and the ETH will move completed to the beacon chain. One fact to be known on this is that you can migrate your ETH from Eth1 to Eth2, but you cannot move it back. It is a unidirectional migration from Proof of Work (PoW) Eth1 to Proof of Stake (PoS) Eth2 model.
Considering the locked state of your ETH will imply that you will need to run two clients on Ethereum blockchain - a Validator client on Eth2 to confirm transactions and new blocks created and a client on Eth1 to monitor and confirm the new staked deposits. This will cover the transition period between Eth1 and Eth2 (if it will be fully done, it remains to know) and help with keeping the speed of Eth2 without adding additional validation layers. Hopefully this bring some light upon how ETH is migrating from Eth1 to Eth2 and the logistics of it.
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