Remark 89.7.8. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. Then there is an induced morphism $\widehat{\varphi }: \widehat{\mathcal{F}} \to \widehat{\mathcal{G}}$ of categories cofibered in groupoids over $\widehat{\mathcal{C}}_\Lambda$. It sends an object $\xi = (R, \xi _ n, f_ n)$ of $\widehat{\mathcal{F}}$ to $(R, \varphi (\xi _ n), \varphi (f_ n))$, and it sends a morphism $(a_0 : R \to S, a_ n : \xi _ n \to \eta _ n)$ between objects $\xi$ and $\eta$ of $\widehat{\mathcal{F}}$ to $(a_0 : R \to S, \varphi (a_ n) : \varphi (\xi _ n) \to \varphi (\eta _ n))$. Finally, if $t : \varphi \to \varphi '$ is a $2$-morphism between $1$-morphisms $\varphi , \varphi ': \mathcal{F} \to \mathcal{G}$ of categories cofibred in groupoids, then we obtain a $2$-morphism $\widehat{t} : \widehat{\varphi } \to \widehat{\varphi }'$. Namely, for $\xi = (R, \xi _ n, f_ n)$ as above we set $\widehat{t}_\xi = (t_{\varphi (\xi _ n)})$. Hence completion defines a functor between $2$-categories

$\widehat{~ } : \text{Cof}(\mathcal{C}_\Lambda ) \longrightarrow \text{Cof}(\widehat{\mathcal{C}}_\Lambda )$

from the $2$-category of categories cofibred in groupoids over $\mathcal{C}_\Lambda$ to the $2$-category of categories cofibred in groupoids over $\widehat{\mathcal{C}}_\Lambda$.

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