The hyperreals contain infinitesimals that are not standard reals, so transferred proofs may track structural isomorphisms rather than truths about the intended real-number system.

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Intended real-number system(as used in philosophy of mathematics): The specific number system that mathematicians originally had in mind to study—in this case, the ordinary real numbers without infinitesimals.
Standard reals(contrasted with non-standard reals): The ordinary real numbers you learn about in regular math class—numbers like 1, 2.5, π, and √2 that can be located precisely on a number line.
Structural isomorphisms(as used in mathematical logic and philosophy of mathematics): Two different systems that have exactly the same shape or pattern, even if they're made of different pieces—like how two differently sized maps can show the same relationships between cities.
Transferred proofs(as used in mathematics and philosophy of mathematics): Mathematical arguments or proofs that are moved from one number system to another and still work, even if they weren't originally designed for the new system.

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