I am currently working on a exercise sheet about categories. There are two exercises:
In the first parts I have to show that the vertical composition and the horizontal composition of two natural transformations define again a natural transformation.
The relevant exercises for my question are:
1) Let $C$ and $D$ be small categories. Show that the following data define a category $Fun(C, D)$: Objects are morphisms from $C$ to $D$, and the morphisms from $F$ to $F'$ are natural transformations. Composition of natural transformations is defined as the vertical composition of natural transformations.
2) Now let $C$ be a small category. Show that the category $Fun(C, C)$ is a strict monoidal category with monoidal product given by the composition of functors and horizontal composition of natural transformations. The monoidal unit is given by the identity functor.
I thought that I proofed everything just fine (used the definition and showed the axioms, pretty easy calculations, especially in the first one).
Unfortunately I don't see where I need the precondition that $C$ is a small category. What is the problem if it is not? Where do I have to use it?
If $C$ is a large category, then a functor $F:C\to D $ cannot be represented by a set but by a proper class only, hence it cannot be an element of the class of functors $\operatorname {Fun}(C,D) $ (because a proper class cannot be an element of any class).
On the ther hand if $C,D $ are both small categories, then $\operatorname {Fun}(C,D) $ is a small category as well.