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Added:
Specifically, a continuous functor $C\to Set$ is a right adjoint functor if and only if it is representable, in which case the left adjoint functor $Set\to C$ sends the singleton set to the representing object
I see now that, originating with the creation of the entry back in 2011, it has a line starting with the words
As representable functors are ubiquitous,…
This does not make sense to me, neither the claim itself nor the suggestion that it implies the statement that follows. I wonder what was really meant here. But it looks like just deleting these words would not take anything away from the paragraphs that follow. (?)
I changed the opening of the first sentence under “Related facts” a little so that it ties in better with the phrase that follows (the earlier version, about the ubiquity of representable functors, seemingly echoes similar statements about the ubiquity of (the concept of) adjoint functors – see the quotations of Mac Lane given in Categories for the Working Mathematician).
Okay, I think my confusion came from the fact that you need $C$ to be cocomplete in order to be able to say that $F : C \to Set$ has a left adjoint iff it is representable, and I assumed that the representable functor theorem followed from the adjoint functor theorem and this equivalence; but the representable functor theorem actually stands on its own (without the assumption that $C$ is cocomplete), and then in order to deduce the adjoint functor theorem from that you only need $D(x, Fy)$ to be representable, so there’s no copowering nonsense.
I think the “specifically” paragraph should be rewritten now, but I am not sure how.
I don’t think there’s any “nonsense”.
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