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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?

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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.

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  • $\begingroup$ A slight improvement of this result only asks $C$ to be small; in that case, $Fun(C,D)$ is the same size of $D$ (so large if $D$ is large), but still "legitimate", in that each hom-class in $Fun(C,D)$ is a set. This is, however, no big deal at the beginning of your studies in category theory. Unfortunately, and especially in old books, CT is taught in a way that forces many novices to familiarize with size issues too soon in their apprenticeship. $\endgroup$ – Fosco Loregian Sep 23 '18 at 13:18

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