isColimitCoconeLeftOpOfCone — Mathlib

Lean4

/-- Turn a limit of `F : J ⥤ Cᵒᵖ` into a colimit of `F.leftOp : Jᵒᵖ ⥤ C`. -/
@[simps]
def isColimitCoconeLeftOpOfCone (F : J ⥤ Cᵒᵖ) {c : Cone F} (hc : IsLimit c) : IsColimit (coconeLeftOpOfCone c)
    where
  desc s := (hc.lift (coneOfCoconeLeftOp s)).unop
  fac s
    j :=
    Quiver.Hom.op_inj <| by
      simp only [coconeLeftOpOfCone_ι_app, op_comp, Quiver.Hom.op_unop, IsLimit.fac, coneOfCoconeLeftOp_π_app, op_unop]
  uniq s m
    w := by
    refine Quiver.Hom.op_inj (hc.hom_ext fun j => Quiver.Hom.unop_inj ?_)
    simpa only [Quiver.Hom.op_unop, IsLimit.fac, coneOfCoconeLeftOp_π_app] using w (op j)

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