The Stacks project

Let $F: \mathcal{C} \to \textit{Sets}$ be a functor. We can think of a set as a discrete category, i.e., as a groupoid with only identity morphisms. Then the construction (9) associates to $F$ a category cofibered in sets. This defines a fully faithful embedding of the category of functors $\mathcal{C} \to \textit{Sets}$ to the category of categories cofibered in groupoids over $\mathcal{C}$. We identify the category of functors with its image under this embedding. Hence if $F : \mathcal{C} \to \textit{Sets}$ is a functor, we denote the associated category cofibered in sets also by $F$; and if $\varphi : F \to G$ is a morphism of functors, we denote still by $\varphi$ the corresponding morphism of categories cofibered in sets, and vice-versa. See Categories, Section 4.38.