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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.

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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 $x$.

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  • $\begingroup$ Borceux is not working with Grothendieck universes in this reference. In fact, in Vol.1 for the "classical" Yoneda Lemma he does not forget to say that we need the category small to have the naturality in $F$. $\endgroup$ – Sov Mar 10 '18 at 17:13
  • $\begingroup$ But I think I made a mistake in my edit because the first Yoneda lemma tell us that all the natural transformations form a set, so I can define an additive structure on it pointwise. Thanks for your answer. $\endgroup$ – Sov Mar 10 '18 at 17:14
  • $\begingroup$ You can define the additive structure just fine even if it is a proper class. The definition will itself be a schema rather than an actual definition in the language of ZFC, but you can still prove all the properties you want to be true as theorem schemas. $\endgroup$ – Eric Wofsey Mar 10 '18 at 17:18
  • $\begingroup$ Yes but I think I'm working in ZFC (sorry I'm not really familiar with those notions). Because when he proves that the category $Add(\mathcal{C},\mathcal{D})$ of additive functors between two preadditive categories is again preadditive, he asks for the category $\mathcal{C}$ to be small. $\endgroup$ – Sov Mar 10 '18 at 17:24
  • $\begingroup$ This is necessary if you want $Add(\mathcal{C},\mathcal{D})$ to literally be a category that you can define as a class. But you can still prove all the properties involved in stating that it is additive if $\mathcal{C}$ is large; they are just theorem schemas (separate theorems for each additive functor $\mathcal{C}\to\mathcal{D}$). You would only run into real trouble if you tried to talk about arbitrary functors from $Add(\mathcal{C},\mathcal{D})$ to another category. $\endgroup$ – Eric Wofsey Mar 10 '18 at 17:30

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