Post Closed as "not a real question" by François G. Dorais, Charles Siegel, Gjergji Zaimi, Yemon Choi, José Figueroa-O'Farrill

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Do we have infinite infinitely many forgetful functors for every pair of categories for which it can those forgetful functors be constructed?

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Forgetful functors.

Do we have infinite forgetful functors for every pair of categories for which it can be constructed?