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

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

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

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

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.

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