Let's use Skolem's paradox to build the category of all sets!

I'm starting to understand category theory, but there is a problem.

There are various foundations of category theory. For example:

I'm happy with none of them, because I want to study category theory within ZFC.

Skolem's paradox (which is not a paradox) says that set theories like ZFC have countable models. So, let's fix a countable model $M$. Then we can define "the category of all sets" $\mathbf{Set}$ whose objects are in $M$. Of course, $M$ doesn't contain all sets, but it can behave as if it does.

In this way, we can rigorously study category theory within ZFC. However, there may be some problems in this approach that I failed to know. So please let me know whether this approach is ok or not.

What you want is answered in this paper.

Feferman, S. and Kreisel, G., Set-theoretical foundations of category theory, Reports of the Midwest Category Seminar III, 201-247

They use the Löwenheim-Skolem construction to develop a theory which is equiconsistent with ZFC and able to formalize category theory.

PS: ZFC + two universes is much more convenient, and while being not equiconsistent with ZFC, experience shows that "concrete" results obtained with this theory are always provable in ZFC (usually because of the reflection principle). But take this with a grain of salt since this only refers to an informal meaning of "concrete statement".