We assume ZFC+U.
A *category* is an ordered pair $(\operatorname{Ob} \mathcal{C},\operatorname{Mor} \mathcal{C},\operatorname{dom},\operatorname{codom},e,∘)$ of sets (not classes) and maps satifying some conditions.

Let $\mathbb{U}$ be a Grothendieck universe.
An element of $\mathbb{U}$ is called a *$\mathbb{U}$-set*.
A set is called *$\mathbb{U}$-small* if it is isomorphic to a $\mathbb{U}$-set.
In the following, we suppose that $\mathbb{N} \in \mathbb{U}$.

In SGA4, a category $\mathcal{C}$ is called *$\mathbb{U}$-small*
if $(\operatorname{Ob} \mathcal{C},\operatorname{Mor} \mathcal{C},\operatorname{dom},\operatorname{codom},e,∘)$ is $\mathbb{U}$-small
as a set (if my understanding is correct).
However, I don't see this definition working well.
For any set $a$ and $b$,
an ordered pair $(a,b)$ is always $\mathbb{U}$-small
since $(a,b)=\{\{a\},\{a,b\} \}$ is a set consisting of exactly two elements,
which is isomophic to $2:=\{\emptyset,\{\emptyset\}\} \in \mathbb{U}$.
Thus, $\mathbb{U}$-smallness imposes nothing on categories.
In particular, it is not equivalent to $\operatorname{Ob} \mathcal{C}$ and $\operatorname{Mor} \mathcal{C}$ are $\mathbb{U}$-small.

I think I am mistaken somewhere, where is it?

notbijection-invariant. $\endgroup$4more comments