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If there is an isomorphism $\mathrm{Hom}(y C, P) \cong P (C)$ natural in $P$ and $C$, then by restricting to the representable functors one obtains an automorphism of the identity functor of the category $C$ comes from. There are categories for which the identity functor has a non-trivial automorphism group, e.g. $\mathbf{Ab}$, and also categories for which the identity functor has a trivial automorphism group, e.g. $\mathbf{Set}$.