File: Mathlib/Order/Category/LinOrd.lean

full_name: LinOrd
code: /-- The category of linear orders. -/
def LinOrd 
goal: Type (u_1 + 1)

full_name: LinOrd.instParentProjectionPartialOrderLinearOrderToPartialOrder
code: instance : BundledHom.ParentProjection @LinearOrder.toPartialOrder 
goal: CategoryTheory.BundledHom.ParentProjection @LinearOrder.toPartialOrder

full_name: LinOrd.instConcreteCategory
code: instance : ConcreteCategory LinOrd 
goal: CategoryTheory.ConcreteCategory LinOrd

full_name: LinOrd.instCoeSortType
code: instance : CoeSort LinOrd Type* 
goal: CoeSort LinOrd (Type u_1)

full_name: LinOrd.of
code: /-- Construct a bundled `LinOrd` from the underlying type and typeclass. -/
def of (α : Type*) [LinearOrder α] : LinOrd 
goal: LinOrd

full_name: LinOrd.coe_of
code: @[simp]
theorem coe_of (α : Type*) [LinearOrder α] : ↥(of α) = α 
goal: ↑(LinOrd.of α) = α

full_name: LinOrd.instInhabited
code: instance : Inhabited LinOrd 
goal: Inhabited LinOrd

full_name: LinOrd.instLinearOrderα
code: instance (α : LinOrd) : LinearOrder α 
goal: LinearOrder ↑α

full_name: LinOrd.hasForgetToLat
code: instance hasForgetToLat : HasForget₂ LinOrd Lat 
goal: CategoryTheory.HasForget₂ LinOrd Lat

full_name: LinOrd.Iso.mk_inv
code: /-- Constructs an equivalence between linear orders from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : LinOrd.{u}} (e : α ≃o β) : α ≅ β 
goal: (LinOrd.Iso.mk e).inv = ↑e.symm

full_name: LinOrd.Iso.mk
code: /-- Constructs an equivalence between linear orders from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : LinOrd.{u}} (e : α ≃o β) : α ≅ β 
goal: α ≅ β

full_name: LinOrd.Iso.mk_hom
code: /-- Constructs an equivalence between linear orders from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : LinOrd.{u}} (e : α ≃o β) : α ≅ β 
goal: (LinOrd.Iso.mk e).hom = ↑e

full_name: LinOrd.dual
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : LinOrd ⥤ LinOrd 
goal: LinOrd ⥤ LinOrd

full_name: LinOrd.dual_map
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : LinOrd ⥤ LinOrd 
goal: ∀ {X Y : LinOrd} (a : ↑X →o ↑Y), LinOrd.dual.map a = OrderHom.dual a

full_name: LinOrd.dual_obj
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : LinOrd ⥤ LinOrd 
goal: LinOrd.dual.obj X = LinOrd.of (↑X)ᵒᵈ

full_name: LinOrd.dualEquiv_inverse
code: /-- The equivalence between `LinOrd` and itself induced by `OrderDual` both ways. -/
@[simps functor inverse]
def dualEquiv : LinOrd ≌ LinOrd 
goal: LinOrd.dualEquiv.inverse = LinOrd.dual

full_name: LinOrd.dualEquiv_functor
code: /-- The equivalence between `LinOrd` and itself induced by `OrderDual` both ways. -/
@[simps functor inverse]
def dualEquiv : LinOrd ≌ LinOrd 
goal: LinOrd.dualEquiv.functor = LinOrd.dual

full_name: LinOrd.dualEquiv
code: /-- The equivalence between `LinOrd` and itself induced by `OrderDual` both ways. -/
@[simps functor inverse]
def dualEquiv : LinOrd ≌ LinOrd 
goal: LinOrd ≌ LinOrd

full_name: linOrd_dual_comp_forget_to_Lat
code: theorem linOrd_dual_comp_forget_to_Lat :
    LinOrd.dual ⋙ forget₂ LinOrd Lat = forget₂ LinOrd Lat ⋙ Lat.dual 
goal: LinOrd.dual ⋙ CategoryTheory.forget₂ LinOrd Lat = CategoryTheory.forget₂ LinOrd Lat ⋙ Lat.dual