When reasoning about arbitrary (large) categories, it's a good idea to use something more than ZFC - maybe you like Grothendieck universes (https://en.wikipedia.org/wiki/Grothendieck_universe), or some class theory such as NBG plus global choice (https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory). ZFC alone doesn't let us do very much. So my understanding is that, in general category theory, it is not unusual to assume a somewhat stronger foundational theory than ZFC.
Now, the way I see it, you have two primary worries: are these stronger foundations consistent, and - if so - are they true?
As far as consistency goes, these stronger foundations are only very slightly stronger, generally. As it turns out, passing to a stronger foundational theory doesn't really take very much effort: for instance, NBG + global choice is consevative over ZFC, meaning that it won't prove any new statements about sets that ZFC didn't already. Similarly, although Grothendieck universes do add some strength to ZFC, they don't add very much (as measured by the large cardinal hierarchy, see ). So if you're comfortable with the consistency of ZFC, then (a) you're comfortable with the consistency of NBG and (b) you shouldn't be too uncomfortable about the consistency of Grothendieck universes.
As to whether these stronger foundations are true - well, that gets into deeper waters. (For one thing, how do you know ZFC itself is "true," and what does "true" mean in this context, anyways?) I'm a formalist (usually), so I don't tend to spend too much time with this question, but there are plenty of places where the truth of large cardinals is defended. (As for NBG + global choice, I'm actually not aware of any strong partisans for its truth over mere NBG.)
This is all leaving aside the question of whether the passage to stronger foundations is even necessary for "ordinary" mathematics, even "ordinary" category theory - I believe the answer is a strong "no," but someone more knowledgeable should address that.
You'll notice that I have a pragmatic attitude towards foundations - "use what works." In general, I think that even ZFC is not to be taken for granted. In my opinion, all theorems that need any "real" set theory (don't ask me to define that :P) should be viewed in the context of the set theory used to prove them. Here there's no difference for me between, say, Borel determinacy, which can be proved in ZFC, but not much less(!), and the result you quote, which can be proved in ZFC + a bit more, but not ZFC alone. If I'm in a context where we're assuming a background theory strong enough to prove them, I can breezily say that they're true - but if I'm being really careful, I should at least tell you some natural sufficient axioms. We might disagree about the truth of those axioms - and hence about the truth of the theorems - but we won't disagree about the purely formal claim, "Such-and-such axioms prove such-and-such theorem."