Lemma 77.23.1. Assumptions and notation as in Lemmas 77.20.2 and 77.20.3. The vertical composition of

$\xymatrix@C=15pc{ \mathcal{S}_{R \times _{s, U, t} R} \ruppertwocell ^{\pi \circ s \circ \text{pr}_1 = \pi \circ s \circ c}{\ \ \ \ \ \ \alpha \star \text{id}_{\text{pr}_1}} \ar[r]_(.3){\pi \circ t \circ \text{pr}_1 = \pi \circ s \circ \text{pr}_0} \rlowertwocell _{\pi \circ t \circ \text{pr}_0 = \pi \circ t \circ c}{\ \ \ \ \ \ \alpha \star \text{id}_{\text{pr}_0}} & [U/R] }$

is the $2$-morphism $\alpha \star \text{id}_ c$. In a formula $\alpha \star \text{id}_ c = (\alpha \star \text{id}_{\text{pr}_0}) \circ (\alpha \star \text{id}_{\text{pr}_1})$.

Proof. We make two remarks:

1. The formula $\alpha \star \text{id}_ c = (\alpha \star \text{id}_{\text{pr}_0}) \circ (\alpha \star \text{id}_{\text{pr}_1})$ only makes sense if you realize the equalities $\pi \circ s \circ \text{pr}_1 = \pi \circ s \circ c$, $\pi \circ t \circ \text{pr}_1 = \pi \circ s \circ \text{pr}_0$, and $\pi \circ t \circ \text{pr}_0 = \pi \circ t \circ c$. Namely, the second one implies the vertical composition $\circ$ makes sense, and the other two guarantee the two sides of the formula are $2$-morphisms with the same source and target.

2. The reason the lemma holds is that composition in the category fibred in groupoids $[U/_{\! p}R]$ associated to the presheaf in groupoids (77.20.0.1) comes from the composition law $c : R \times _{s, U, t} R \to R$.

We omit the proof of the lemma. $\square$

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