descFun — Mathlib

Lean4
/-- The function from the colimit monoid to the cone point of any other cocone. -/
def descFun (s : Cocone F) : ColimitType F → s.pt :=
  by
  fapply Quot.lift
  · exact descFunLift F s
  · intro x y r
    induction r with
    | refl x => rfl
    | symm x y _ h => exact h.symm
    | trans x y z _ _ h₁ h₂ => exact h₁.trans h₂
    | map j j' f x => exact s.w_apply f x
    | mul j x y => exact map_mul (s.ι.app j).hom x y
    | one j => exact map_one (s.ι.app j).hom
    | mul_1 x x' y _ h => exact congr_arg (· * _) h
    | mul_2 x y y' _ h => exact congr_arg (_ * ·) h
    | mul_assoc x y z => exact mul_assoc _ _ _
    | one_mul x => exact one_mul _
    | mul_one x => exact mul_one _
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