English
The main compatibility statement of opensImagePreimageMap_app holds: the red and blue arrows commute in the overarching gluing diagram.
Русский
Основное тождество opensImagePreimageMap_app: красные и синие стрелки в большом диаграммном склейивании commute.
LaTeX
$$$\\text{opensImagePreimageMap\_app}(i,j,k,U)$$$
Lean4
/-- The red and the blue arrows in  commute. -/
@[simp, reassoc]
theorem snd_invApp_t_app (i j k : D.J) (U : Opens (pullback (D.f i j) (D.f i k)).carrier) :
(π₂⁻¹ i, j, k) U ≫ (D.t k i).c.app _ =
(D.t' k i j).c.app _ ≫
(π₁⁻¹ k, j, i) (unop _) ≫ (D.V (k, i)).presheaf.map (eqToHom (D.snd_invApp_t_app' i j k U).choose.symm) :=
by
have e := (D.snd_invApp_t_app' i j k U).choose_spec
replace e := reassoc_of% e
rw [← e]
simp [eqToHom_map]