File: Mathlib/CategoryTheory/Limits/Shapes/Pullback/Square.lean

full_name: CategoryTheory.Square.pullbackCone
code: /-- The pullback cone attached to a commutative square. -/
abbrev pullbackCone : PullbackCone sq.f₂₄ sq.f₃₄ 
goal: CategoryTheory.Limits.PullbackCone sq.f₂₄ sq.f₃₄

full_name: CategoryTheory.Square.pushoutCocone
code: /-- The pushout cocone attached to a commutative square. -/
abbrev pushoutCocone : PushoutCocone sq.f₁₂ sq.f₁₃ 
goal: CategoryTheory.Limits.PushoutCocone sq.f₁₂ sq.f₁₃

full_name: CategoryTheory.Square.IsPullback
code: /-- The condition that a commutative square is a pullback square. -/
protected def IsPullback : Prop 
goal: Prop

full_name: CategoryTheory.Square.IsPushout
code: /-- The condition that a commutative square is a pushout square. -/
protected def IsPushout : Prop 
goal: Prop

full_name: CategoryTheory.Square.isPullback_iff
code: lemma isPullback_iff :
    sq.IsPullback ↔ Nonempty (IsLimit sq.pullbackCone) 
goal: sq.IsPullback ↔ Nonempty (CategoryTheory.Limits.IsLimit sq.pullbackCone)

full_name: CategoryTheory.Square.isPushout_iff
code: lemma isPushout_iff :
    sq.IsPushout ↔ Nonempty (IsColimit sq.pushoutCocone) 
goal: sq.IsPushout ↔ Nonempty (CategoryTheory.Limits.IsColimit sq.pushoutCocone)

full_name: CategoryTheory.Square.IsPullback.mk
code: lemma IsPullback.mk (h : IsLimit sq.pullbackCone) : sq.IsPullback 
goal: sq.IsPullback

full_name: CategoryTheory.Square.IsPushout.mk
code: lemma IsPushout.mk (h : IsColimit sq.pushoutCocone) : sq.IsPushout 
goal: sq.IsPushout

full_name: CategoryTheory.Square.IsPullback.isLimit
code: /-- If a commutative square `sq` is a pullback square,
then `sq.pullbackCone` is limit. -/
noncomputable def IsPullback.isLimit (h : sq.IsPullback) :
    IsLimit sq.pullbackCone 
goal: CategoryTheory.Limits.IsLimit sq.pullbackCone

full_name: CategoryTheory.Square.IsPushout.isColimit
code: /-- If a commutative square `sq` is a pushout square,
then `sq.pushoutCocone` is colimit. -/
noncomputable def IsPushout.isColimit (h : sq.IsPushout) :
    IsColimit sq.pushoutCocone 
goal: CategoryTheory.Limits.IsColimit sq.pushoutCocone

full_name: CategoryTheory.Square.IsPullback.of_iso
code: lemma IsPullback.of_iso {sq₁ sq₂ : Square C} (h : sq₁.IsPullback)
    (e : sq₁ ≅ sq₂) : sq₂.IsPullback 
goal: sq₂.IsPullback

full_name: CategoryTheory.Square.IsPullback.iff_of_iso
code: lemma IsPullback.iff_of_iso {sq₁ sq₂ : Square C} (e : sq₁ ≅ sq₂) :
    sq₁.IsPullback ↔ sq₂.IsPullback 
goal: sq₁.IsPullback ↔ sq₂.IsPullback

full_name: CategoryTheory.Square.IsPushout.of_iso
code: lemma IsPushout.of_iso {sq₁ sq₂ : Square C} (h : sq₁.IsPushout)
    (e : sq₁ ≅ sq₂) : sq₂.IsPushout 
goal: sq₂.IsPushout

full_name: CategoryTheory.Square.IsPushout.iff_of_iso
code: lemma IsPushout.iff_of_iso {sq₁ sq₂ : Square C} (e : sq₁ ≅ sq₂) :
    sq₁.IsPushout ↔ sq₂.IsPushout 
goal: sq₁.IsPushout ↔ sq₂.IsPushout

full_name: CategoryTheory.Square.IsPushout.op
code: lemma IsPushout.op {sq : Square C} (h : sq.IsPushout) : sq.op.IsPullback 
goal: sq.op.IsPullback

full_name: CategoryTheory.Square.IsPushout.unop
code: lemma IsPushout.unop {sq : Square Cᵒᵖ} (h : sq.IsPushout) : sq.unop.IsPullback 
goal: sq.unop.IsPullback

full_name: CategoryTheory.Square.IsPullback.op
code: lemma IsPullback.op {sq : Square C} (h : sq.IsPullback) : sq.op.IsPushout 
goal: sq.op.IsPushout

full_name: CategoryTheory.Square.IsPullback.unop
code: lemma IsPullback.unop {sq : Square Cᵒᵖ} (h : sq.IsPullback) : sq.unop.IsPushout 
goal: sq.unop.IsPushout