4.1.1. Warmup: locally small family fibrations [000G]

An ordinary category $E$ is called locally small when for any $x,y\in E$ the collection of morphisms $x\to y$ is a set. This property of $E$ can be rephrased in terms of its family fibration (Section 2.7 [0006]) $\FAM{E}$ over $\SET$ as follows.

Consider an index set $I\in \SET$ and two families $u,v\in C^I$. We may define an $I$-indexed collection $[u,v]\Sub{i\in I}$ consisting (pointwise) of all the morphisms $u\Sub{i}\to v\Sub{i}$ in $C$:

If $C$ is locally small, $[u,v]\Sub{i\in I}$ is in fact a family of sets for any $I\in\SET$ as each $[u,v]\Sub{i}$ is a set. Conversely, if $[u,v]\Sub{i\in I}$ is a family of sets for any $I\in \SET$, then $C$ is locally small as we may consider in particular the case that $I=\mathbf{1}$.