In this section we discuss how to go between categories cofibred in groupoids over $\mathcal{C}_\Lambda $ to categories cofibred in groupoids over $\widehat{\mathcal{C}}_\Lambda $ and vice versa.
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.
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 $R/\mathfrak m_ R^{n + 1} \to R/\mathfrak m_ R^ n$.
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.
The category of formal objects comes with a functor $\widehat{p}: \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda $ which sends an object $(R, \xi _ n, f_ n)$ to $R$ and a morphism $(R, \xi _ n, f_ n) \to (S, \eta _ n, g_ n)$ to the map $R \to S$.
Lemma 90.7.2. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda $ be a category cofibered in groupoids. Then $\widehat{p} : \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda $ is a category cofibered in groupoids.
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
\[ \xymatrix{ \xi _{n + 1} \ar[d] \ar[r]_{f_ n} & \xi _ n \ar[d] \\ \eta _{n + 1} \ar[r]^{g_ n} & \eta _ n } \]
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 $h_ n \circ c_{n + 1} = c_ n \circ g_ n$).
$\square$
Definition 90.7.3. Let $p : \mathcal{F} \to \mathcal{C}_\Lambda $ be a category cofibered in groupoids. The category cofibered in groupoids $\widehat{p} : \widehat{\mathcal F} \to \widehat{\mathcal{C}}_\Lambda $ is called the completion of $\mathcal{F}$.
If $\mathcal{F}$ is a category cofibered in groupoids over $\mathcal C_\Lambda $, we have defined $\widehat{\mathcal{F}}(R)$ for $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda )$ in terms of the filtration of $R$ by powers of its maximal ideal. But suppose $\mathcal{I} = (I_ n)$ is a filtration of $R$ by ideals inducing the $\mathfrak {m}_ R$-adic topology. We define $\widehat{\mathcal{F}}_\mathcal {I}(R)$ to be the category with the following objects and morphisms:
An object is a collection $(\xi _ n, f_ n)_{n \in \mathbf{N}}$ of objects $\xi _ n$ of $\mathcal{F}(R/I_ n)$ and morphisms $f_ n : \xi _{n + 1} \to \xi _ n$ lying over the projections $R/I_{n + 1} \to R/I_ n$.
A morphism $a : (\xi _ n, f_ n) \to (\eta _ n, g_ n)$ consists of a collection $a_ n : \xi _ n \to \eta _ n$ of morphisms in $\mathcal{F}(R/I_ n)$, 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.
Lemma 90.7.4. In the situation above, $\widehat{\mathcal{F}}_\mathcal {I}(R)$ is equivalent to the category $\widehat{\mathcal{F}}(R)$.
Proof.
An equivalence $\widehat{\mathcal{F}}_\mathcal {I}(R) \to \widehat{\mathcal{F}}(R)$ can be defined as follows. For each $n$, let $m(n)$ be the least $m$ that $I_ m \subset \mathfrak m_ R^ n$. Given an object $(\xi _ n, f_ n)$ of $\widehat{\mathcal{F}}_\mathcal {I}(R)$, let $\eta _ n$ be the pushforward of $\xi _{m(n)}$ along $R/I_{m(n)} \to R/\mathfrak m_ R^ n$. Let $g_ n : \eta _{n + 1} \to \eta _ n$ be the unique morphism of $\mathcal{F}$ lying over $R/\mathfrak m_ R^{n + 1} \to R/\mathfrak m_ R^ n$ such that
\[ \xymatrix{ \xi _{m(n + 1)} \ar[rrr]_{f_{m(n)} \circ \ldots \circ f_{m(n + 1) - 1}} \ar[d] & & & \xi _{m(n)} \ar[d] \\ \eta _{n + 1} \ar[rrr]^{g_ n} & & & \eta _ n } \]
commutes (existence and uniqueness is guaranteed by the axioms of a cofibred category). The functor $\widehat{\mathcal{F}}_\mathcal {I}(R) \to \widehat{\mathcal{F}}(R)$ sends $(\xi _ n, f_ n)$ to $(R, \eta _ n, g_ n)$. We omit the verification that this is indeed an equivalence of categories.
$\square$
Having said this, the equivalence $\Phi : \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \mathcal{G}|_{\mathcal{C}_\Lambda }, \mathcal{F}) \to \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }(\mathcal{G}, \widehat{\mathcal{F}})$ sends a morphism $\varphi : \mathcal{G}|_{\mathcal{C}_\Lambda } \to \mathcal{F}$ to
\[ \mathcal{G} \to \widehat{\mathcal{G}|_{\mathcal{C}_\Lambda }} \xrightarrow {\widehat{\varphi }} \widehat{\mathcal{F}} \]
There is a quasi-inverse $\Psi : \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_\Lambda }( \mathcal{G}, \widehat{\mathcal{F}}) \to \mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}_\Lambda }( \mathcal{G}|_{\mathcal{C}_\Lambda }, \mathcal{F})$ to $\Phi $ which sends $\psi : \mathcal{G} \to \widehat{\mathcal{F}}$ to
\[ \mathcal{G}|_{\mathcal{C}_\Lambda } \xrightarrow {\psi |_{\mathcal{C}_\Lambda }} \widehat{\mathcal{F}}|_{\mathcal{C}_\Lambda } \to \mathcal{F}. \]
We omit the verification that $\Phi $ and $\Psi $ are quasi-inverse. We also do not address functoriality of $\Phi $ (because it would lead into 3-category territory which we want to avoid at all cost).
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