Remark 90.7.13 (06SL)—The Stacks project

Remark 90.7.13. Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda $. Let $\xi = (R, \xi _ n, f_ n)$ and $\eta = (S, \eta _ n, g_ n)$ be formal objects of $\mathcal{F}$. Let $a = (a_ n) : \xi \to \eta $ be a morphism of formal objects, i.e., a morphism of $\widehat{\mathcal{F}}$. Let $f = \widehat{p}(a) = a_0 : R \to S$ be the projection of $a$ in $\widehat{\mathcal{C}}_\Lambda $. Then we obtain a $2$-commutative diagram

\[ \xymatrix{ \underline{R}|_{\mathcal{C}_\Lambda } \ar[rd]_{\underline{\xi }} & & \underline{S}|_{\mathcal{C}_\Lambda } \ar[ll]^ f \ar[ld]^{\underline{\eta }} \\ & \mathcal{F} } \]

where $\underline{\xi }$ and $\underline{\eta }$ are the morphisms constructed in Remark 90.7.12. To see this let $\alpha : S \to A$ be an object of $\underline{S}|_{\mathcal{C}_\Lambda }$ (see loc. cit.). Let $m \in \mathbf{N}$ be minimal such that $\mathfrak m_ A^ m = 0$. We get a commutative diagram

\[ \xymatrix{ R \ar[d]^ f \ar[r] & R/\mathfrak m_ R^ m \ar[d]_{f_ m} \ar[rd]^{\beta _ m} \\ S \ar[r] & S/\mathfrak m_ S^ m \ar[r]^{\alpha _ m} & A } \]

such that the bottom arrows compose to give $\alpha $. Then $\underline{\eta }(\alpha ) = \alpha _{m, *}\eta _ m$ and $\underline{\xi }(\alpha \circ f) = \beta _{m, *}\xi _ m$. The morphism $a_ m : \xi _ m \to \eta _ m$ lies over $f_ m$ hence we obtain a canonical morphism

\[ \underline{\xi }(\alpha \circ f) = \beta _{m, *}\xi _ m \longrightarrow \underline{\eta }(\alpha ) = \alpha _{m, *}\eta _ m \]

lying over $\text{id}_ A$ such that

\[ \xymatrix{ \xi _ m \ar[r] \ar[d]^{a_ m} & \beta _{m, *}\xi _ m \ar[d] \\ \eta _ m \ar[r] & \alpha _{m, *}\eta _ m } \]

commutes by the axioms of a category cofibred in groupoids. This defines a transformation of functors $\underline{\xi } \circ f \to \underline{\eta }$ which witnesses the 2-commutativity of the first diagram of this remark.

The Stacks project

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