The Stacks project

Proof. Let $R \to S$ be a ring map in $\widehat{\mathcal{C}}_\Lambda $. Let $(R, \xi _ n, f_ n)$ be an object of $\widehat{\mathcal{F}}$. For each $n$ choose a pushforward $\xi _ n \to \eta _ n$ of $\xi _ n$ along $R/\mathfrak m_ R^ n \to S/\mathfrak m_ S^ n$. For each $n$ there exists a unique morphism $g_ n : \eta _{n + 1} \to \eta _ n$ in $\mathcal{F}$ lying over $S/\mathfrak m_ S^{n + 1} \to S/\mathfrak m_ S^ n$ such that

commutes (by the first axiom of a category cofibred in groupoids). Hence we obtain a morphism $(R, \xi _ n, f_ n) \to (S, \eta _ n, g_ n)$ lying over $R \to S$, i.e., the first axiom of a category cofibred in groupoids holds for $\widehat{\mathcal{F}}$. To see the second axiom suppose that we have morphisms $a : (R, \xi _ n, f_ n) \to (S, \eta _ n, g_ n)$ and $b : (R, \xi _ n, f_ n) \to (T, \theta _ n, h_ n)$ in $\widehat{\mathcal{F}}$ and a morphism $c_0 : S \to T$ in $\widehat{\mathcal{C}}_\Lambda $ such that $c_0 \circ a_0 = b_0$. By the second axiom of a category cofibred in groupoids for $\mathcal{F}$ we obtain unique maps $c_ n : \eta _ n \to \theta _ n$ lying over $S/\mathfrak m_ S^ n \to T/\mathfrak m_ T^ n$ such that $c_ n \circ a_ n = b_ n$. Setting $c = (c_ n)_{n \geq 0}$ gives the desired morphism $c : (S, \eta _ n, g_ n) \to (T, \theta _ n, h_ n)$ in $\widehat{\mathcal{F}}$ (we omit the verification that $h_ n \circ c_{n + 1} = c_ n \circ g_ n$). $\square$