-- Module: Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits
Goal:
X Y : LocallyRingedSpace
f g : X ⟶ Y
U : Opens (↑(↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace : TopCat) : Type v)
s : (↑((coequalizer (Hom.toShHom f) (Hom.toShHom g)).presheaf.obj (op U)) : Type v)
H : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ Eq.{v + 1} (α := Opens (↑(↑X.toPresheafedSpace : TopCat) : Type v))
    (X.toRingedSpace.basicOpen (U :=
      (Opens.map f.base).obj
        ((Opens.map
                (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)))) :
              Opens (↑(↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace : TopCat) : Type v) ⥤
                Opens (↑(↑Y.toPresheafedSpace : TopCat) : Type v)).1
          U))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{v}) ↦ (↑R : Type v) →+* (↑S : Type v)) (CC :=
            CommRingCat.carrier) (X :=
            Y.presheaf.obj
              (op
                ((Opens.map
                        (PresheafedSpace.Hom.base
                          (InducedCategory.Hom.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)))) :
                      Opens (↑(↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace : TopCat) : Type v) ⥤
                        Opens (↑(↑Y.toPresheafedSpace : TopCat) : Type v)).1
                  U)))
            (f.c.app
              (op
                ((Opens.map
                        (PresheafedSpace.Hom.base
                          (InducedCategory.Hom.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)))) :
                      Opens (↑(↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace : TopCat) : Type v) ⥤
                        Opens (↑(↑Y.toPresheafedSpace : TopCat) : Type v)).1
                  U))) :
          (a :
              (↑(Y.presheaf.obj (op ((Opens.map (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base).1 U))) :
                Type v)) →
            (↑(X.presheaf.obj
                  (op
                    ((Opens.map f.base).obj
                      ((Opens.map (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base).1 U)))) :
              Type v))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{v}) ↦ (↑R : Type v) →+* (↑S : Type v)) (CC :=
                CommRingCat.carrier) (Y :=
                Y.presheaf.obj
                  (op
                    (Functor.obj
                      (Opens.map
                        (PresheafedSpace.Hom.base
                          (InducedCategory.Hom.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)))))
                      U)))
                (NatTrans.app
                  (PresheafedSpace.Hom.c (InducedCategory.Hom.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g))))
                  (op U)) :
              (a : (↑((coequalizer (Hom.toShHom f) (Hom.toShHom g)).presheaf.obj (op U)) : Type v)) →
                (↑(Y.presheaf.obj (op ((Opens.map (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base).obj U))) :
                  Type v))
            s :
          (↑(Y.presheaf.obj
                (op
                  (Functor.obj
                    (Opens.map
                      (PresheafedSpace.Hom.base
                        (InducedCategory.Hom.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)))))
                    U))) :
            Type v))))
    (X.toRingedSpace.basicOpen (U :=
      (Opens.map g.base).obj
        ((Opens.map
                (PresheafedSpace.Hom.base (InducedCategory.Hom.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)))) :
              Opens (↑(↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace : TopCat) : Type v) ⥤
                Opens (↑(↑Y.toPresheafedSpace : TopCat) : Type v)).1
          U))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{v}) ↦ (↑R : Type v) →+* (↑S : Type v)) (CC :=
            CommRingCat.carrier) (X :=
            Y.presheaf.obj
              (op
                ((Opens.map
                        (PresheafedSpace.Hom.base
                          (InducedCategory.Hom.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)))) :
                      Opens (↑(↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace : TopCat) : Type v) ⥤
                        Opens (↑(↑Y.toPresheafedSpace : TopCat) : Type v)).1
                  U)))
            (g.c.app
              (op
                ((Opens.map
                        (PresheafedSpace.Hom.base
                          (InducedCategory.Hom.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)))) :
                      Opens (↑(↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace : TopCat) : Type v) ⥤
                        Opens (↑(↑Y.toPresheafedSpace : TopCat) : Type v)).1
                  U))) :
          (a :
              (↑(Y.presheaf.obj (op ((Opens.map (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base).1 U))) :
                Type v)) →
            (↑(X.presheaf.obj
                  (op
                    ((Opens.map g.base).obj
                      ((Opens.map (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base).1 U)))) :
              Type v))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{v}) ↦ (↑R : Type v) →+* (↑S : Type v)) (CC :=
                CommRingCat.carrier) (Y :=
                Y.presheaf.obj
                  (op
                    (Functor.obj
                      (Opens.map
                        (PresheafedSpace.Hom.base
                          (InducedCategory.Hom.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)))))
                      U)))
                (NatTrans.app
                  (PresheafedSpace.Hom.c (InducedCategory.Hom.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g))))
                  (op U)) :
              (a : (↑((coequalizer (Hom.toShHom f) (Hom.toShHom g)).presheaf.obj (op U)) : Type v)) →
                (↑(Y.presheaf.obj (op ((Opens.map (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base).obj U))) :
                  Type v))
            s :
          (↑(Y.presheaf.obj
                (op
                  (Functor.obj
                    (Opens.map
                      (PresheafedSpace.Hom.base
                        (InducedCategory.Hom.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)))))
                    U))) :
            Type v))))

Tactic:
rw [← ConcreteCategory.comp_apply, ← PresheafedSpace.comp_c_app,
      ← CommRingCat.comp_apply, ← PresheafedSpace.comp_c_app]

Premises used: CategoryTheory.ConcreteCategory.comp_apply, AlgebraicGeometry.PresheafedSpace.comp_c_app, CommRingCat.comp_apply