File: Mathlib/CategoryTheory/FinCategory/AsType.lean

full_name: CategoryTheory.FinCategory.ObjAsType
code: /-- A FinCategory `α` is equivalent to a category with objects in `Type`. -/
--@[nolint unused_arguments]
abbrev ObjAsType : Type 
goal: Type

full_name: CategoryTheory.FinCategory.instFintypeHomObjAsType
code: instance {i j : ObjAsType α} : Fintype (i ⟶ j) 
goal: Fintype (i ⟶ j)

full_name: CategoryTheory.FinCategory.objAsTypeEquiv
code: /-- The constructed category is indeed equivalent to `α`. -/
noncomputable def objAsTypeEquiv : ObjAsType α ≌ α 
goal: CategoryTheory.FinCategory.ObjAsType α ≌ α

full_name: CategoryTheory.FinCategory.AsType
code: /-- A FinCategory `α` is equivalent to a fin_category with in `Type`. -/
--@[nolint unused_arguments]
abbrev AsType : Type 
goal: Type

full_name: CategoryTheory.FinCategory.categoryAsType
code: @[simps (config := .lemmasOnly) id comp]
noncomputable instance categoryAsType : SmallCategory (AsType α) 
goal: CategoryTheory.SmallCategory (CategoryTheory.FinCategory.AsType α)

full_name: CategoryTheory.FinCategory.categoryAsType_comp
code: @[simps (config := .lemmasOnly) id comp]
noncomputable instance categoryAsType : SmallCategory (AsType α) 
goal: ∀ {X Y Z : CategoryTheory.FinCategory.AsType α} (f : X ⟶ Y) (g : Y ⟶ Z),
  CategoryTheory.CategoryStruct.comp f g =
    (Fintype.equivFin (X ⟶ Z))
      (CategoryTheory.CategoryStruct.comp ((Fintype.equivFin (X ⟶ Y)).symm f) ((Fintype.equivFin (Y ⟶ Z)).symm g))

full_name: CategoryTheory.FinCategory.categoryAsType_id
code: @[simps (config := .lemmasOnly) id comp]
noncomputable instance categoryAsType : SmallCategory (AsType α) 
goal: CategoryTheory.CategoryStruct.id i = (Fintype.equivFin (i ⟶ i)) (CategoryTheory.CategoryStruct.id i)

full_name: CategoryTheory.FinCategory.asTypeToObjAsType
code: /-- The "identity" functor from `AsType α` to `ObjAsType α`. -/
@[simps]
noncomputable def asTypeToObjAsType : AsType α ⥤ ObjAsType α 
goal: CategoryTheory.Functor (CategoryTheory.FinCategory.AsType α) (CategoryTheory.FinCategory.ObjAsType α)

full_name: CategoryTheory.FinCategory.asTypeToObjAsType_map
code: /-- The "identity" functor from `AsType α` to `ObjAsType α`. -/
@[simps]
noncomputable def asTypeToObjAsType : AsType α ⥤ ObjAsType α 
goal: (CategoryTheory.FinCategory.asTypeToObjAsType α).map a = (Fintype.equivFin (X ⟶ Y)).symm a

full_name: CategoryTheory.FinCategory.asTypeToObjAsType_obj
code: /-- The "identity" functor from `AsType α` to `ObjAsType α`. -/
@[simps]
noncomputable def asTypeToObjAsType : AsType α ⥤ ObjAsType α 
goal: (CategoryTheory.FinCategory.asTypeToObjAsType α).obj a = id a

full_name: CategoryTheory.FinCategory.objAsTypeToAsType
code: /-- The "identity" functor from `ObjAsType α` to `AsType α`. -/
@[simps]
noncomputable def objAsTypeToAsType : ObjAsType α ⥤ AsType α 
goal: CategoryTheory.Functor (CategoryTheory.FinCategory.ObjAsType α) (CategoryTheory.FinCategory.AsType α)

full_name: CategoryTheory.FinCategory.objAsTypeToAsType_map
code: /-- The "identity" functor from `ObjAsType α` to `AsType α`. -/
@[simps]
noncomputable def objAsTypeToAsType : ObjAsType α ⥤ AsType α 
goal: (CategoryTheory.FinCategory.objAsTypeToAsType α).map a = (Fintype.equivFin (X ⟶ Y)) a

full_name: CategoryTheory.FinCategory.objAsTypeToAsType_obj
code: /-- The "identity" functor from `ObjAsType α` to `AsType α`. -/
@[simps]
noncomputable def objAsTypeToAsType : ObjAsType α ⥤ AsType α 
goal: (CategoryTheory.FinCategory.objAsTypeToAsType α).obj a = id a

full_name: CategoryTheory.FinCategory.asTypeEquivObjAsType
code: /-- The constructed category (`AsType α`) is equivalent to `ObjAsType α`. -/
noncomputable def asTypeEquivObjAsType : AsType α ≌ ObjAsType α 
goal: CategoryTheory.FinCategory.AsType α ≌ CategoryTheory.FinCategory.ObjAsType α

full_name: CategoryTheory.FinCategory.asTypeFinCategory
code: noncomputable instance asTypeFinCategory : FinCategory (AsType α) 
goal: CategoryTheory.FinCategory (CategoryTheory.FinCategory.AsType α)

full_name: CategoryTheory.FinCategory.equivAsType
code: /-- The constructed category (`ObjAsType α`) is indeed equivalent to `α`. -/
noncomputable def equivAsType : AsType α ≌ α 
goal: CategoryTheory.FinCategory.AsType α ≌ α