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I'm reading Borceux-Handbook of categorical algebra 1, p.11.

The author assumes a category $\mathscr{C}$ to be a collection consists of "classes" of $Obj(\mathscr{C})$ and $Mor(\mathscr{C})$ where $Mor(\mathscr{C})$ consists of sets $Hom(A,B)$, where the term class means that in NBG set theory, and assumes other usual category axioms. (Composition and identity axioms) (So that this definition is stronger than the definition of locally small category in naive category theory)

Let $F:\mathscr{C}\rightarrow \textbf{Set}$ be a covariant functor. The author states that the collection of natural transformations $Nat(\mathscr{C}(A,-),F)$ is a class, but why is it so?

When stating the Yoneda lemma, he simply constructs $Nat(\mathscr{C}(A,-),F)$, but this cannot be constructed unless one shows that this is indeed a class under NBG. Am I correct?

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    I would argue that $Nat(\mathscr{C}(A,-),F)$ can be obtained by separation as a subclass of $\prod_{B\in Obj(\mathscr{C})} \mathbf{Set}(\mathscr{C}(A,B),F(B))$, the latter being a class, as a product of a family of sets indexed by a class. I am no expert though, so I may well be wrong. – 2017-01-18

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Of course it is not the case that natural transformations $\mathscr{C}(A,-)\Rightarrow F$ form a class because natural transformations are functions between classes $Obj(\mathscr{C})\to Mor(\mathbf{Set})$, i.e. certain subclasses of the class of pairs $Obj(\mathscr{C})\times Mor(\mathbf{Set})$, hence cannot be members of a class.

The Yoneda lemma asserts, however, that there is a (natural) bijective correspondence between natural transformations $\mathscr{C}(A,-)\Rightarrow F$ and elements of the small class $F(A)$. Accordingly, we define $Nat(\mathscr C(A,-),F)$ to be the class $F(A)$.

More generally (and this applies also in the case where the category $\mathscr C$ is not locally small), you should think of $\mathscr C(A,-)$ as a class-valued copresheaf (which is roughly a family of classes indexed by the objects of $\mathscr C$ equipped with a "functorial" action of the morphisms of $\mathscr C$). Then the Yoneda lemma asserts that given another class-valued copresheaf $F$, there is a (natural) bijective correspondence between morphisms of copresheaves $\mathscr C(A,-)\Rightarrow F$ and elements of the class $F(A)$. Thus the Yoneda lemma allows you to realize as a class the a priori "metaclass" of morphisms from a representable class-valued copresheaf on a (not necessarily locally small) category.

What is usually called the Yoneda lemma is the exact same argument but with $F$ restricted to being a class-valued presheaf valued in small classes, i.e. a functor to $\mathbf{Set}$ instead. But there is no difference in the proofs and you might as well use the more general statement to define the class of natural transformations; then it's by definition that it's a set for a set-valued copresheaf (i.e. functor to $\mathbf{Set}$).

More generally still, this is a special case of the Yoneda lemma for fibrations, which should be thought of as well-behaved category-valued (weak) copresheaves on the $\mathscr C$ given its natural structure (via the arrow category $\mathscr C^\to$) as a category internal to the "metacategory" of (possibly large) categories. The Yoneda lemma for fibrations is slightly different and gives a realization as a category of the "metacategory" of morphisms only up to equivalence because fibrations are weak copresheaves.