File: Mathlib/Order/Category/Preord.lean

full_name: Preord
code: /-- The category of preorders. -/
def Preord 
goal: Type (u_1 + 1)

full_name: Preord.instBundledHomPreorderOrderHom
code: instance : BundledHom @OrderHom 
goal: CategoryTheory.BundledHom OrderHom

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

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

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

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

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

full_name: Preord.instPreorderα
code: instance (α : Preord) : Preorder α 
goal: Preorder ↑α

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

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

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

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

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

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

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

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

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

full_name: preordToCat_map
code: /-- The embedding of `Preord` into `Cat`.
-/
@[simps]
def preordToCat : Preord.{u} ⥤ Cat 
goal: ∀ {X Y : Preord} (f : X ⟶ Y), preordToCat.map f = ⋯.functor

full_name: preordToCat_obj
code: /-- The embedding of `Preord` into `Cat`.
-/
@[simps]
def preordToCat : Preord.{u} ⥤ Cat 
goal: preordToCat.obj X = CategoryTheory.Cat.of ↑X

full_name: preordToCat
code: /-- The embedding of `Preord` into `Cat`.
-/
@[simps]
def preordToCat : Preord.{u} ⥤ Cat 
goal: Preord ⥤ CategoryTheory.Cat

full_name: instFaithfulPreordCatPreordToCat
code: instance : preordToCat.{u}.Faithful 
goal: preordToCat.Faithful

full_name: instFullPreordCatPreordToCat
code: instance : preordToCat.{u}.Full 
goal: preordToCat.Full