The representable functor theorem states that:
Dually, the corepresentable functor theorem states that:
As representable functors are closely connected to adjoint functors, this theorem is essentially equivalent to the adjoint functor theorem and to theorems guaranteeing the existence of limits.
Specifically, assuming that $C$ is copowered over Set (in particular, this is true if $C$ is cocomplete), a 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, and more generally a set $X$ to the copower of $X$ with the representing object.
Bodo Pareigis, Thm. 1 on p. 109 in: Categories and Functors, Pure and Applied Mathematics 39, Academic Press (1970) [doi:10.5282/ubm/epub.7244, pdf]
Saunders MacLane, p. 118 (2nd ed: 130) in: Categories for the Working Mathematician, Graduate texts in mathematics 5, Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]
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