The hom-set form of an adjunction needs no size hypothesis

Form a one-object category $\mathcal{O}$ whose endomorphism class is $\mathbf{O}\mathbf{r}\mathbf{d}$, whose identity is $0$, and whose composition is ordinal addition. The zero clause in [F2] gives $\alpha +0=\alpha$ directly. The other identity law $0+\alpha =\alpha$ needs all three clauses and transfinite induction on $\alpha$: the zero clause gives $0+0=0$; the successor clause gives $0+(\beta +1)=(0+\beta )+1=\beta +1$ from the inductive hypothesis; and for a nonzero limit $\lambda$ the limit clause gives $0+\lambda ={\bigcup }_{\beta <\lambda }(0+\beta )={\bigcup }_{\beta <\lambda }\beta =\lambda$. Associativity is [F3].

Its only hom-class is the proper class $\mathbf{O}\mathbf{r}\mathbf{d}$ by [F4], so $\mathcal{O}$ is not locally small by [F1]. Consequently $\mathcal{O}(\ast ,\ast )$ is not a hom-set and cannot be an object of $\mathbf{S}\mathbf{e}\mathbf{t}$.

Thus a Set-valued hom-set bijection is not even meaningful in this example, whereas [L1] explains why unit-counit data remain the size-free formulation. The statement is false.