$ \newcommand\cc{\mathcal{C}} \newcommand\set{\mathsf{Set}} \newcommand\sbcat{\ensuremath{\mathit{Category}}}

$

Definition 1  (Definition of set-based categories)

Let $\cc$ be a small category and let $U:\cc\to\set$ be a functor. We

say that $\cc$ is set-based ($\sbcat$ for short) with respect to $U$ if the pair $(\cc,U)$ satisfy the following:

Notice that the condition (Sb:b) implies that the functor $U$ preserves image--factorizations: