isLimitConeLeftOpOfCocone — Mathlib

Lean4

/-- Turn a colimit for `F : J ⥤ Cᵒᵖ` into a limit for `F.leftOp : Jᵒᵖ ⥤ C`. -/
@[simps]
def isLimitConeLeftOpOfCocone (F : J ⥤ Cᵒᵖ) {c : Cocone F} (hc : IsColimit c) : IsLimit (coneLeftOpOfCocone c)
    where
  lift s := (hc.desc (coconeOfConeLeftOp s)).unop
  fac s
    j :=
    Quiver.Hom.op_inj <| by
      simp only [coneLeftOpOfCocone_π_app, op_comp, Quiver.Hom.op_unop, IsColimit.fac, coconeOfConeLeftOp_ι_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, IsColimit.fac, coconeOfConeLeftOp_ι_app] using w (op j)

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