Lincoln didn’t support the Corwin Amendment - he only said that he had “no objection”.
splitting hairs
I have no purpose, directly or indirectly, to interfere with the institution of slavery in the States where it exists. I believe I have no lawful right to do so, and I have no inclination to do so.
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And:
I understand a proposed amendment to the Constitution—which amendment, however, I have not seen—has passed Congress, to the effect that the Federal Government shall never interfere with the domestic institutions of the States, including that of persons held to service. To avoid misconstruction of what I have said, I depart from my purpose not to speak of particular amendments so far as to say that, holding such a provision to now be implied constitutional law, I have no objection to its being made express and irrevocable.