4.3. Separators for cartesian fibrations [002E]
Let $E$ be an ordinary category. In general, to compare two morphisms $f,g:x\to y$ in $E$, it is not enough to see if they agree on global points $u:1\to x$, because the behavior of $u,v$ may differ only on generalized elements. In some cases, however, there is a family of objects $\prn{s\Sub{i}}\Sub{i\in I}\in E$ are together adequate for comparing morphisms of $E$ in the sense of Definition 4.3·a [002G] below.
Given an ordinary category $E$, a set-indexed family $\prn{s\Sub{i}}\Sub{i\in I}$ of $E$-objects is called a small separating family for $E$ when, assuming that for all $i\in I$ and all $u:s_i\to x$ we have $u;f=u;g$, we then have $f=g$.
The intuition of Definition 4.3·a [002G] is that to compare two morphisms $f,g:x\to y\in E$, it suffices to check that they behave the same on all $s\Sub{i}$-shaped figures when $\prn{s\Sub{i}}\Sub{i\in I}$ is a separating family for $E$.
In the category of sets, to compare two morphisms it is enough to check that they agree on global points. This means that the unary family $\brc{\ObjTerm{\SET}}$ is a separator for $\SET$, a property referred to more generally as well-pointedness.
We will now generalize Definition 4.3·a [002G] to the case of a cartesian fibration.
Let $\bar{s}$ be a displayed object in a cartesian fibration $E$ over $B$. A pair of displayed morphisms $f,g:\bar{x}\to \bar{y}\in E$ are said to agree on $\bar{s}$-figures when for any $\bar{s}$-figure $h : \bar{z}\to \bar{x}$ in the sense of Definition 2.9·b [002K], we have $h;f = h;g : \bar{z}\to \bar{y}$.
Let $E$ be a cartesian fibration over $B$ such that $B$ has binary products. A displayed object $\bar{s}\in E\Sub{s}$ is said to be a small separator for $E$ when any two vertical maps $f,g:\bar{u}\to\Sub{\Idn{x}}\bar{v}\in E\Sub{x}$ are equal when they are agree on $\bar{s}$-figures in the sense of Definition 4.3·c [002I].