File: Mathlib/CategoryTheory/Shift/Opposite.lean

full_name: CategoryTheory.HasShift.mkShiftCoreOp
code: /-- Construction of the naive shift on the opposite category of a category `C`:
the shiftfunctor by `n` is `(shiftFunctor C n).op`. -/
noncomputable def mkShiftCoreOp : ShiftMkCore Cᵒᵖ A 
goal: CategoryTheory.ShiftMkCore Cᵒᵖ A

full_name: CategoryTheory.OppositeShift
code: /-- The category `OppositeShift C A` is the opposite category `Cᵒᵖ` equipped
with the naive shift: `shiftFunctor (OppositeShift C A) n` is `(shiftFunctor C n).op`. -/
@[nolint unusedArguments]
def OppositeShift (A : Type*) [AddMonoid A] [HasShift C A] 
goal: Type u_1

full_name: CategoryTheory.instCategoryOppositeShift
code: instance : Category (OppositeShift C A) 
goal: CategoryTheory.Category.{u_3, u_1} (CategoryTheory.OppositeShift C A)

full_name: CategoryTheory.instHasShiftOppositeShift
code: noncomputable instance : HasShift (OppositeShift C A) A 
goal: CategoryTheory.HasShift (CategoryTheory.OppositeShift C A) A

full_name: CategoryTheory.instHasZeroObjectOppositeShift
code: instance [HasZeroObject C] : HasZeroObject (OppositeShift C A) 
goal: CategoryTheory.Limits.HasZeroObject (CategoryTheory.OppositeShift C A)

full_name: CategoryTheory.instPreadditiveOppositeShift
code: instance [Preadditive C] : Preadditive (OppositeShift C A) 
goal: CategoryTheory.Preadditive (CategoryTheory.OppositeShift C A)

full_name: CategoryTheory.instAdditiveOppositeShiftShiftFunctor
code: instance [Preadditive C] (n : A) [(shiftFunctor C n).Additive] :
    (shiftFunctor (OppositeShift C A) n).Additive 
goal: (CategoryTheory.shiftFunctor (CategoryTheory.OppositeShift C A) n).Additive

full_name: CategoryTheory.oppositeShiftFunctorZero_inv_app
code: lemma oppositeShiftFunctorZero_inv_app (X : OppositeShift C A) :
    (shiftFunctorZero (OppositeShift C A) A).inv.app X =
      ((shiftFunctorZero C A).hom.app X.unop).op 
goal: (CategoryTheory.shiftFunctorZero (CategoryTheory.OppositeShift C A) A).inv.app X =
  ((CategoryTheory.shiftFunctorZero C A).hom.app (Opposite.unop X)).op

full_name: CategoryTheory.oppositeShiftFunctorZero_hom_app
code: lemma oppositeShiftFunctorZero_hom_app (X : OppositeShift C A) :
    (shiftFunctorZero (OppositeShift C A) A).hom.app X =
      ((shiftFunctorZero C A).inv.app X.unop).op 
goal: (CategoryTheory.shiftFunctorZero (CategoryTheory.OppositeShift C A) A).hom.app X =
  ((CategoryTheory.shiftFunctorZero C A).inv.app (Opposite.unop X)).op

full_name: CategoryTheory.oppositeShiftFunctorAdd_inv_app
code: lemma oppositeShiftFunctorAdd_inv_app :
    (shiftFunctorAdd (OppositeShift C A) a b).inv.app X =
      ((shiftFunctorAdd C a b).hom.app X.unop).op 
goal: (CategoryTheory.shiftFunctorAdd (CategoryTheory.OppositeShift C A) a b).inv.app X =
  ((CategoryTheory.shiftFunctorAdd C a b).hom.app (Opposite.unop X)).op

full_name: CategoryTheory.oppositeShiftFunctorAdd_hom_app
code: lemma oppositeShiftFunctorAdd_hom_app :
    (shiftFunctorAdd (OppositeShift C A) a b).hom.app X =
      ((shiftFunctorAdd C a b).inv.app X.unop).op 
goal: (CategoryTheory.shiftFunctorAdd (CategoryTheory.OppositeShift C A) a b).hom.app X =
  ((CategoryTheory.shiftFunctorAdd C a b).inv.app (Opposite.unop X)).op

full_name: CategoryTheory.oppositeShiftFunctorAdd'_inv_app
code: lemma oppositeShiftFunctorAdd'_inv_app :
    (shiftFunctorAdd' (OppositeShift C A) a b c h).inv.app X =
      ((shiftFunctorAdd' C a b c h).hom.app X.unop).op 
goal: (CategoryTheory.shiftFunctorAdd' (CategoryTheory.OppositeShift C A) a b c h).inv.app X =
  ((CategoryTheory.shiftFunctorAdd' C a b c h).hom.app (Opposite.unop X)).op

full_name: CategoryTheory.oppositeShiftFunctorAdd'_hom_app
code: lemma oppositeShiftFunctorAdd'_hom_app :
    (shiftFunctorAdd' (OppositeShift C A) a b c h).hom.app X =
      ((shiftFunctorAdd' C a b c h).inv.app X.unop).op 
goal: (CategoryTheory.shiftFunctorAdd' (CategoryTheory.OppositeShift C A) a b c h).hom.app X =
  ((CategoryTheory.shiftFunctorAdd' C a b c h).inv.app (Opposite.unop X)).op