The Stacks project

Remark 4.2.11. Suppose that $\mathcal{A}$ is a category. A functor $F$ from $\mathcal{A}$ to $\textit{Sets}$ is a mathematical object (i.e., it is a set not a class or a formula of set theory, see Sets, Section 3.2) even though the category of sets is “big”. Namely, the range of $F$ on objects will be a set $F(\mathop{\mathrm{Ob}}\nolimits (\mathcal{A}))$ and then we may think of $F$ as a functor between $\mathcal{A}$ and the full subcategory of the category of sets whose objects are elements of $F(\mathop{\mathrm{Ob}}\nolimits (\mathcal{A}))$.