The Stacks project

Definition 90.7.1. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $. The category $\widehat{\mathcal{F}}$ of formal objects of $\mathcal{F}$ is the category with the following objects and morphisms.

1. A formal object $\xi = (R, \xi _ n, f_ n)$ of $\mathcal{F}$ consists of an object $R$ of $\widehat{\mathcal{C}}_\Lambda $, and a collection indexed by $n \in \mathbf{N}$ of objects $\xi _ n$ of $\mathcal{F}(R/\mathfrak m_ R^ n)$ and morphisms $f_ n : \xi _{n + 1} \to \xi _ n$ lying over the projection $R/\mathfrak m_ R^{n + 1} \to R/\mathfrak m_ R^ n$.

2. Let $\xi = (R, \xi _ n, f_ n)$ and $\eta = (S, \eta _ n, g_ n)$ be formal objects of $\mathcal{F}$. A morphism $a : \xi \to \eta $ of formal objects consists of a map $a_0 : R \to S$ in $\widehat{\mathcal{C}}_\Lambda $ and a collection $a_ n : \xi _ n \to \eta _ n$ of morphisms of $\mathcal{F}$ lying over $R/\mathfrak m_ R^ n \to S/\mathfrak m_ S^ n$, such that for every $n$ the diagram

   \[ \xymatrix{ \xi _{n + 1} \ar[r]_{f_ n} \ar[d]_{a_{n + 1}} & \xi _ n \ar[d]^{a_ n} \\ \eta _{n + 1} \ar[r]^{g_ n} & \eta _ n } \]

   commutes.