Russell's Paradox (https://en.wikipedia.org/wiki/Russell%27s_paradox) means you cannot have a set-of-all-sets. For the category of sets and functions between sets, that means you have to use the weaker idea of a 'collection'.
A positive side-effect of this avoidance of set axioms in the definition of a category is that we can more easily define categories with very different ideas of membership. For example, a category where membership is a real number (like in fuzzy logic). Categories with this idea of 'generalised membership' (plus a few extra bits and bobs) are called 'toposes', and can be used in much the same way as sets are. Importantly, we can relate different categories like this together using the same terminology and concepts and so more easily define ways to join classical code with fuzzy logic, or with neural networks (or anything else which satisfies the topos axioms) and know they're not insane.
Incidentally, you can define the axioms of a topos (and sets are a topos) in purely category theory terms. That is why some (and I'm one of them) view categories as a more useful foundation than set theory (where attempts to define the converse are much more messy).
A positive side-effect of this avoidance of set axioms in the definition of a category is that we can more easily define categories with very different ideas of membership. For example, a category where membership is a real number (like in fuzzy logic). Categories with this idea of 'generalised membership' (plus a few extra bits and bobs) are called 'toposes', and can be used in much the same way as sets are. Importantly, we can relate different categories like this together using the same terminology and concepts and so more easily define ways to join classical code with fuzzy logic, or with neural networks (or anything else which satisfies the topos axioms) and know they're not insane.
Incidentally, you can define the axioms of a topos (and sets are a topos) in purely category theory terms. That is why some (and I'm one of them) view categories as a more useful foundation than set theory (where attempts to define the converse are much more messy).