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Its like this. Your funds belong to your seed. Thats it. Whatever wallet you enter it into will bring up your blockchain address and all the tokens in the account. The seed is your private key access (and control) of that wallet/account/blockchain public address.
Note: The moment you enter your cold storage seed phrase in a hot wallet you have rendered that seed phrase hot, as in its now a hot wallet phrase only because it has left the cold storage onto hot storage. It is no longer in cold storage only. Losing its secure, (hard wallet) cold-stored only existence.