File: Mathlib/CategoryTheory/Monoidal/Internal/Module.lean

full_name: ModuleCat.MonModuleEquivalenceAlgebra.Ring_of_Mon_
code: instance Ring_of_Mon_ (A : Mon_ (ModuleCat.{u} R)) : Ring A.X 
goal: Ring ↑A.X

full_name: ModuleCat.MonModuleEquivalenceAlgebra.Algebra_of_Mon_
code: instance Algebra_of_Mon_ (A : Mon_ (ModuleCat.{u} R)) : Algebra R A.X 
goal: Algebra R ↑A.X

full_name: ModuleCat.MonModuleEquivalenceAlgebra.algebraMap
code: @[simp]
theorem algebraMap (A : Mon_ (ModuleCat.{u} R)) (r : R) : algebraMap R A.X r = A.one r 
goal: (algebraMap R ↑A.X) r = A.one r

full_name: ModuleCat.MonModuleEquivalenceAlgebra.functor
code: /-- Converting a monoid object in `ModuleCat R` to a bundled algebra.
-/
@[simps!]
def functor : Mon_ (ModuleCat.{u} R) ⥤ AlgebraCat R 
goal: Mon_ (ModuleCat R) ⥤ AlgebraCat R

full_name: ModuleCat.MonModuleEquivalenceAlgebra.functor_obj_carrier
code: /-- Converting a monoid object in `ModuleCat R` to a bundled algebra.
-/
@[simps!]
def functor : Mon_ (ModuleCat.{u} R) ⥤ AlgebraCat R 
goal: ↑(ModuleCat.MonModuleEquivalenceAlgebra.functor.obj A) = ↑A.X

full_name: ModuleCat.MonModuleEquivalenceAlgebra.functor_map_apply
code: /-- Converting a monoid object in `ModuleCat R` to a bundled algebra.
-/
@[simps!]
def functor : Mon_ (ModuleCat.{u} R) ⥤ AlgebraCat R 
goal: (ModuleCat.MonModuleEquivalenceAlgebra.functor.map f) a = f.hom a

full_name: ModuleCat.MonModuleEquivalenceAlgebra.inverseObj
code: /-- Converting a bundled algebra to a monoid object in `ModuleCat R`.
-/
@[simps]
def inverseObj (A : AlgebraCat.{u} R) : Mon_ (ModuleCat.{u} R) 
goal: Mon_ (ModuleCat R)

full_name: ModuleCat.MonModuleEquivalenceAlgebra.inverseObj_one
code: /-- Converting a bundled algebra to a monoid object in `ModuleCat R`.
-/
@[simps]
def inverseObj (A : AlgebraCat.{u} R) : Mon_ (ModuleCat.{u} R) 
goal: (ModuleCat.MonModuleEquivalenceAlgebra.inverseObj A).one = Algebra.linearMap R ↑A

full_name: ModuleCat.MonModuleEquivalenceAlgebra.inverseObj_mul
code: /-- Converting a bundled algebra to a monoid object in `ModuleCat R`.
-/
@[simps]
def inverseObj (A : AlgebraCat.{u} R) : Mon_ (ModuleCat.{u} R) 
goal: (ModuleCat.MonModuleEquivalenceAlgebra.inverseObj A).mul = LinearMap.mul' R ↑A

full_name: ModuleCat.MonModuleEquivalenceAlgebra.inverseObj_X
code: /-- Converting a bundled algebra to a monoid object in `ModuleCat R`.
-/
@[simps]
def inverseObj (A : AlgebraCat.{u} R) : Mon_ (ModuleCat.{u} R) 
goal: (ModuleCat.MonModuleEquivalenceAlgebra.inverseObj A).X = ModuleCat.of R ↑A

full_name: ModuleCat.MonModuleEquivalenceAlgebra.inverse_map_hom
code: /-- Converting a bundled algebra to a monoid object in `ModuleCat R`.
-/
@[simps]
def inverse : AlgebraCat.{u} R ⥤ Mon_ (ModuleCat.{u} R) 
goal: ∀ {X Y : AlgebraCat R} (f : X ⟶ Y), (ModuleCat.MonModuleEquivalenceAlgebra.inverse.map f).hom = AlgHom.toLinearMap f

full_name: ModuleCat.MonModuleEquivalenceAlgebra.inverse
code: /-- Converting a bundled algebra to a monoid object in `ModuleCat R`.
-/
@[simps]
def inverse : AlgebraCat.{u} R ⥤ Mon_ (ModuleCat.{u} R) 
goal: AlgebraCat R ⥤ Mon_ (ModuleCat R)

full_name: ModuleCat.MonModuleEquivalenceAlgebra.inverse_obj
code: /-- Converting a bundled algebra to a monoid object in `ModuleCat R`.
-/
@[simps]
def inverse : AlgebraCat.{u} R ⥤ Mon_ (ModuleCat.{u} R) 
goal: ModuleCat.MonModuleEquivalenceAlgebra.inverse.obj A = ModuleCat.MonModuleEquivalenceAlgebra.inverseObj A

full_name: ModuleCat.monModuleEquivalenceAlgebra
code: /-- The category of internal monoid objects in `ModuleCat R`
is equivalent to the category of "native" bundled `R`-algebras.
-/
def monModuleEquivalenceAlgebra : Mon_ (ModuleCat.{u} R) ≌ AlgebraCat R 
goal: Mon_ (ModuleCat R) ≌ AlgebraCat R

full_name: ModuleCat.monModuleEquivalenceAlgebraForget
code: /-- The equivalence `Mon_ (ModuleCat R) ≌ AlgebraCat R`
is naturally compatible with the forgetful functors to `ModuleCat R`.
-/
def monModuleEquivalenceAlgebraForget :
    MonModuleEquivalenceAlgebra.functor ⋙ forget₂ (AlgebraCat.{u} R) (ModuleCat.{u} R) ≅
      Mon_.forget (ModuleCat.{u} R) 
goal: ModuleCat.MonModuleEquivalenceAlgebra.functor ⋙ CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R) ≅
  Mon_.forget (ModuleCat R)