I'm trying to read a bit up on category theory, but I'm a bit confused about one aspect of the proof of Yoneda's lemma. Suppose we have a locally small category C, a functor [itex]F : C \to \textrm{Set}[/itex] and an object A in C. Now according to Yoneda's lemma there exists a bijection from [itex]Nat(hom(A,-),F)[/itex] to [itex]FA[/itex]. Assuming [itex]Nat(hom(A,-),F)[/itex] is a class I can easily construct an explicit bijection which shows that [itex]Nat(hom(A,-),F)[/itex] is actually a set. However all sources I have looked at take it for granted that [itex]Nat(hom(A,-),F)[/itex] is a class and simply starts by defining a function [itex]\Theta_{F,A} : Nat(hom(A,-),F) \to FA[/itex] and then shows that it's bijective. I'm not convinced that [itex]Nat(hom(A,-),F)[/itex] actually exists and isn't contradictory in some way though. I guess it's something obvious I'm missing as it's always left out, but I would appreciate it if someone would tell me what I'm missing.(adsbygoogle = window.adsbygoogle || []).push({});

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# Why is Nat(hom(A,-),F) a class?

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