The Stacks project

Remark 4.2.16. This is one instance where the same thing does not hold if $\mathcal{A}$ is a “big” category. For example consider functors $\textit{Sets} \to \textit{Sets}$. As we have currently defined it such a functor is a class and not a set. In other words, it is given by a formula in set theory (with some variables equal to specified sets)! It is not a good idea to try to consider all possible formulae of set theory as part of the definition of a mathematical object. The same problem presents itself when considering sheaves on the category of schemes for example. We will come back to this point later.