I'm studying Abelian Categories in F. Borceux "Handbook of Categorical Algebra" Vol.2.
In this reference we can find an additive version of the famous Yoneda Lemma :
Let $\mathcal{C}$ be a preadditive category, $A$ be an object in $\mathcal{C}$ and $F : \mathcal{C} \rightarrow \mathbf{Ab}$ be an additive functor. Then there exist isomorphisms $\Theta_{F,A}$ of abelian groups \begin{align*} \Theta_{F,A} : Nat(Hom_{\mathcal{C}}(A,\_),F) \cong F(A) \end{align*} which are natural both in $F$ and $A$.
My question is : Why did he not suppose that the category $\mathcal{C}$ is small? I think we need it to state the naturality in $F$, like in the original Yoneda lemma.
Edit : Moreover we need it to define an additive structure on natural transformations and then check that $\Theta_{F,A}$ is a group morphism.
You're right that if you want to state this as a theorem in ZFC (where large categories are proper classes), you need $\mathcal{C}$ to be small. It is possible that Borceaux is instead working in the framework of Grothendieck universes or something similar, where actually all categories are assumed to be sets (even "large" ones).
Note though that even in ZFC, you can state and prove this as a theorem schema, where you just have a separate theorem for each different formula that could represent all the proper classes involved. So, for instance, there is no actual class of natural transformations between two functors $\mathcal{C}\to\mathbf{Ab}$, but given two such functors $F$ and $G$ and a natural transformation $T$ between them and one particular element $x$ of $\mathrm{Nat}(\mathrm{Hom}_{\mathcal{C}}(A,\_),F)$, you can prove the commutativity of the diagram saying $\Theta_{-,A}$ is natural for $T$ when evaluated at $x$.