TITLE: When size matters in category theory for the working mathematician
QUESTION [58 upvotes]: 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.

REPLY [25 votes]: The other answers are good, but I would like to point out that Ivan's "uncheatable" lemma can in fact be cheated.  The proof of that lemma (due to Freyd) makes inescapable use of classical logic, and in constructive mathematics it is possible to have a non-poset that is complete for the size of its own set of objects (a complete small category).  It is even possible to have a category "of sets" with this property (e.g. those called "modest sets" in realizability).  Then all Kan extensions into such a category exist, all sheafifications of modest presheaves exist, and presumably all Bousfield localizations of modest spectra exist (although the latter may get you into HoTT water when you try to do it constructively).
About the only thing the category of modest sets lacks is a subobject classifier (it is locally cartesian closed).  So these days I prefer the following argument as the "least cheatable" (calling something "uncheatable" sounds like a challenge) manifestation of size issues in category theory.
Lemma 1: Any endofunctor of a complete small category has a fixed point.
Proof: If $C$ is complete-small, so is the category of $F$-algebras for any endofunctor $F:C\to C$.  But any complete small category has an initial object (by essentially the same argument that any complete meet-semilattice also has all joins), and an initial $F$-algebra is a fixed point of $F$ (by Lambek's lemma).
Lemma 2: If $C$ is an elementary topos, the double powerset functor $X \mapsto \Omega^{\Omega^X}$ has no fixed point.
Proof: By Cantor's diagonalization argument.
Thus, no elementary topos can have all limits of the size of its collection of objects.