isLimitConeOfCoconeLeftOp — Mathlib

Lean4

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

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