2.1. The canonical self-indexing [0003]

[001X] Construction 2.1·a (The canonical self-indexing).

Let $B$ be an ordinary category; there is a canonical displayed category $\SelfIx{B}$ over $B$ given fiberwise by the slices of $B$.

1. For $x\in B$, we define $\SelfIx{B}\Sub{x}$ to be the collection $\Sl{B}{x}$ of pairs $(\bar{x}\in B,p\Sub{x}:\bar{x}\to x)$.
2. For $f : x\to y\in B$, we define $\SelfIx{B}\Sub{f}$ to be the collection of commuting squares in the following configuration:

[001Y] Exercise 2.1·b.

Prove that $\SelfIx{B}$ from [001X] is a cartesian fibration if and only if $B$ has pullbacks.