# Category of functors $[C,D]$ and Grothendieck universes

A set $$U$$ is a universe if

• for any $$x \in U$$ we have $$x \subseteq U$$,

• for any $$x,y \in U$$ we have $$\{x,y\} \in U$$,

• for any $$x \in U$$ we have $$\mathcal{P}(x) \in U$$,

• for any family $$(x_i)_{i \in I}$$ of elements $$x_i$$ of $$U$$ indexed by an element $$I$$ of $$U$$ we have $$\bigcup_{i\in I} x_i \in U$$

• $$\mathbb{N} \in U$$.

Let $$U$$ be a universe. We say that a category $$\mathcal{C}$$ is a $$U$$-small if $$Ob(\mathcal{C}) \in U$$ and $$Mor(\mathcal{C}) \in U.$$ We say that a category $$\mathcal{C}$$ is a $$U$$-category if $$Ob(\mathcal{C}) \subseteq U$$ and each $$Hom_{\mathcal{C}}(X,Y) \in U.$$

Let $$U$$ be a universe, let $$C$$ be a $$U$$-small category and let $$D$$ be a $$U$$-category. Is it true that the category $$[C,D]$$ of functors between these categories is necessarily a $$U$$-category? The result is stated in SGA, but they use a slightly different definition of a $$U$$-category (in particular, the set of objects of their $$U$$-category doesn't have to be a subset of $$U$$).

By SGA:

1). A set $$X$$ is called $$\mathcal{U}$$-small, if there exists a set $$Y$$, such that $$Y\in\mathcal{U}$$ and $$X\cong Y$$;

2). A category $$\mathcal{C}$$ is called $$\mathcal{U}$$-small, if the sets $$\text{Obj}(\mathcal{C})$$ and $$\text{Mor}(\mathcal{C})$$ are $$\mathcal{U}$$-small.

3). A category $$\mathcal{C}$$ is called a $$\mathcal{U}$$-category, if for every $$c_1,c_2\in\text{Obj}(\mathcal{C})$$ the set $$\hom_{\mathcal{C}}(c_1,c_2)$$ is $$\mathcal{U}$$-small.

Indeed, if $$\mathcal{C}$$ is a $$\mathcal{U}$$-small category and $$\mathcal{D}$$ is a $$\mathcal{U}$$-category, then the category of functors from $$\mathcal{C}$$ to $$\mathcal{D}$$ is a $$\mathcal{U}$$-category. The proof is that the set of natural transformations between $$\mathcal{F}\colon\mathcal{C}\to\mathcal{D}$$ and $$\mathcal{G}\colon\mathcal{C}\to\mathcal{D}$$ is isomorphic to a subset of $$\prod_{c\in Obj(\mathcal{C})}\hom_{\mathcal{D}}(\mathcal{F}(c),\mathcal{G}(c))$$, which is $$\mathcal{U}$$-small.

1). A category $$\mathcal{C}$$ is $$\mathcal{U}$$-small, if $$\text{Obj}(\mathcal{C})\in\mathcal{U}$$ and $$\text{Mor}(\mathcal{C})\in\mathcal{U}$$;
2). A category $$\mathcal{C}$$ is a $$\mathcal{U}$$-category, if $$\text{Obj}(\mathcal{C})\subset\mathcal{U}$$ and $$\hom_{\mathcal{C}}(c_1,c_2)\in\mathcal{U}$$ for every $$c_1,c_2\in\text{Obj}(\mathcal{C})$$.
Of course, your definitions are not equivalent to those from SGA. Then the analogous statement about functor categories is wrong or requires very unnatural definitons to be true. For example, if a functor $$\mathcal{F}\colon\mathcal{C}\to\mathcal{D}$$ is a quadruplet $$(\mathcal{C},\mathcal{D},\mathcal{F}_{\text{Obj}},\mathcal{F}_{\text{Mor}})$$, then the category of functors $$\mathcal{D}^{\mathcal{C}}$$ fails to be a $$\mathcal{U}$$-category, because every functor should be an element of $$\mathcal{U}$$, but a quadruplet $$(\mathcal{C},\mathcal{D},\mathcal{F}_{\text{Obj}},\mathcal{F}_{\text{Mor}})$$ is not (simple set-theoretical formal verification). On the other hand, if a functor $$\mathcal{F}\colon\mathcal{C}\to\mathcal{D}$$ is a pair $$(\mathcal{F}_{\text{Obj}},\mathcal{F}_{\text{Mor}})$$ and a natural transformation $$\alpha\colon\mathcal{F}\to\mathcal{G}$$ is a subset of $$\text{Obj}(\mathcal{C})\times\text{Mor}(\mathcal{D})$$ (not a triplet $$(\mathcal{F},\mathcal{G},R_{\alpha})$$, where $$R_{\alpha}\subset \text{Obj}(\mathcal{C})\times\text{Mor}(\mathcal{D})$$), then there may happen a situation where domain and codomain of a natural transformation are not defined. And only if you adjust definitions of category, functor and natural transformation such that the set of natural transformations between $$\mathcal{F}$$ and $$\mathcal{G}$$ will be exactly a subset of $$\prod_{c\in Obj(\mathcal{C})}\hom_{\mathcal{D}}(\mathcal{F}(c),\mathcal{G}(c))$$, then this statement will be true. The reason why authors of SGA have chosen the definitions of $$\mathcal{U}$$-small category and $$\mathcal{U}$$-category with the requirements to corresponding sets only being isomorphic to elements of $$\mathcal{U}$$, is that, I guess, they didn't want to deal with these irrelevant set-theoretical issues.