I think a related question might be this (Set-Theoretic Issues/Categories).

There are many ways in which you can avoid set theoretical paradoxes in dealing with category theory (see for instance Shulman https://arxiv.org/abs/0810.1279).

Some important results in category theory assume some kind of 'smallness' of your category in practice. A very much used result in homological algebra is Freyd-Mitchell embedding theorem:

• Every small abelian category admits an fully faithful exact embedding in a category $R$-mod for a suitable ring $R$.

Now, in everyday usage of this result, the restriction on that the category is small in not important: for instance, if you want to do diagram chasing in a diagram on any category, you can allways restrict your attention to the abelian subcategory generated by the objects and maps on the diagram, and the category will be small.

I am wondering

What are results of category theory, commonly used in the mathematical practice, in which considerations of size are crucial?

Shulman in [op. cit.] gives what I think is an example, Freyd Special Adjoint Functor Theorem: a functor from a complete, locally small, and well-powered category with a cogenerating set to a locally small category has a left adjoint if and only if it preserves small limits.

I would find interesting to see some discussion on this topic.

I think a related question might be this (Set-Theoretic Issues/Categories).

There are many ways in which you can avoid set theoretical paradoxes in dealing with category theory (see for instance Shulman - Set theory for category theory).

Some important results in category theory assume some kind of ‘smallness’ of your category in practice. A very much used result in homological algebra is the Freyd–Mitchell embedding theorem:

• Every small abelian category admits an fully faithful exact embedding in a category $\text{$R$-mod}$ for a suitable ring $R$.

Now, in everyday usage of this result, the restriction that the category is small is not important: for instance, if you want to do diagram chasing in a diagram on any category, you can always restrict your attention to the abelian subcategory generated by the objects and maps on the diagram, and the category will be small.

I am wondering:

What are results of category theory, commonly used in mathematical practice, in which considerations of size are crucial?

Shulman in [op. cit.] gives what I think is an example, the Freyd Special Adjoint Functor Theorem: a functor from a complete, locally small, and well-powered category with a cogenerating set to a locally small category has a left adjoint if and only if it preserves small limits.

I would find interesting to see some discussion on this topic.