Lemma 88.8.8. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a smooth morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. Assume $\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$ is essentially surjective. Then $\varphi : \mathcal{F} \to \mathcal{G}$ and $\widehat{\varphi } : \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}$ are essentially surjective.

Proof. Let $y$ be an object of $\mathcal{G}$ lying over $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_\Lambda )$. Let $y \to y_0$ be a pushforward of $y$ along $A \to k$. By the assumption on essential surjectivity of $\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$ there exist an object $x_0$ of $\mathcal{F}$ lying over $k$ and an isomorphism $y_0 \to \varphi (x_0)$. Smoothness of $\varphi$ implies there exists an object $x$ of $\mathcal{F}$ over $A$ whose image $\varphi (x)$ is isomorphic to $y$. Thus $\varphi : \mathcal{F} \to \mathcal{G}$ is essentially surjective.

Let $\eta = (R, \eta _ n, g_ n)$ be an object of $\widehat{\mathcal{G}}$. We construct an object $\xi$ of $\widehat{\mathcal{F}}$ with an isomorphism $\eta \to \varphi (\xi )$. By the assumption on essential surjectivity of $\varphi : \mathcal{F}(k) \to \mathcal{G}(k)$, there exists a morphism $\eta _1 \to \varphi (\xi _1)$ in $\mathcal{G}(k)$ for some $\xi _1 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. The morphism $\eta _2 \xrightarrow {g_1} \eta _1 \to \varphi (\xi _1)$ lies over the surjective ring map $R/\mathfrak m_ R^2 \to k$, hence by smoothness of $\varphi$ there exists $\xi _2 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(R/\mathfrak m_ R^2))$, a morphism $f_1: \xi _2 \to \xi _1$ lying over $R/\mathfrak m_ R^2 \to k$, and a morphism $\eta _2 \to \varphi (\xi _2)$ such that

$\xymatrix{ \varphi (\xi _2) \ar[r]^{\varphi (f_1)} & \varphi (\xi _{1}) \\ \eta _2 \ar[u] \ar[r]^{g_1} & \eta _1 \ar[u] \\ }$

commutes. Continuing in this way we construct an object $\xi = (R, \xi _ n, f_ n)$ of $\widehat{\mathcal{F}}$ and a morphism $\eta \to \varphi (\xi ) = (R, \varphi (\xi _ n), \varphi (f_ n))$ in $\widehat{\mathcal{G}}(R)$. $\square$

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