Module: Mathlib.CategoryTheory.MorphismProperty.Descent

Goal state before:
pp: case e'_2.h₀
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
P : MorphismProperty C
inst✝³ : (isomorphisms C).DescendsAlong P
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : HasPullbacks C
inst✝ : HasEqualizers C
S T : C
f : T ⟶ S
hf : P f
X Y : Over S
a b : X ⟶ Y
hab : (Over.pullback f).map a = (Over.pullback f).map b
e_1✝ : (pullback X.hom f ⟶ pullback Y.hom f) = (((Over.pullback f).obj X).left ⟶ ((Over.pullback f).obj Y).left)
⊢ pullback.map X.hom f Y.hom f (Over.Hom.left a) (𝟙 T) (𝟙 S) ⋯ ⋯ ≫ pullback.fst Y.hom f =
    Over.Hom.left ((Over.pullback f).map a) ≫ pullback.fst Y.hom f

pp: case h₁
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
P : MorphismProperty C
inst✝³ : (isomorphisms C).DescendsAlong P
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : HasPullbacks C
inst✝ : HasEqualizers C
S T : C
f : T ⟶ S
hf : P f
X Y : Over S
a b : X ⟶ Y
hab : (Over.pullback f).map a = (Over.pullback f).map b
e_1✝ : (pullback X.hom f ⟶ pullback Y.hom f) = (((Over.pullback f).obj X).left ⟶ ((Over.pullback f).obj Y).left)
⊢ pullback.map X.hom f Y.hom f (Over.Hom.left a) (𝟙 T) (𝟙 S) ⋯ ⋯ ≫ pullback.snd Y.hom f =
    Over.Hom.left ((Over.pullback f).map a) ≫ pullback.snd Y.hom f

pp: case e'_3.h₀
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
P : MorphismProperty C
inst✝³ : (isomorphisms C).DescendsAlong P
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : HasPullbacks C
inst✝ : HasEqualizers C
S T : C
f : T ⟶ S
hf : P f
X Y : Over S
a b : X ⟶ Y
hab : (Over.pullback f).map a = (Over.pullback f).map b
e_1✝ : (pullback X.hom f ⟶ pullback Y.hom f) = (((Over.pullback f).obj X).left ⟶ ((Over.pullback f).obj Y).left)
⊢ pullback.map X.hom f Y.hom f (Over.Hom.left b) (𝟙 T) (𝟙 S) ⋯ ⋯ ≫ pullback.fst Y.hom f =
    Over.Hom.left ((Over.pullback f).map b) ≫ pullback.fst Y.hom f

pp: case h₁
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
P : MorphismProperty C
inst✝³ : (isomorphisms C).DescendsAlong P
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : HasPullbacks C
inst✝ : HasEqualizers C
S T : C
f : T ⟶ S
hf : P f
X Y : Over S
a b : X ⟶ Y
hab : (Over.pullback f).map a = (Over.pullback f).map b
e_1✝ : (pullback X.hom f ⟶ pullback Y.hom f) = (((Over.pullback f).obj X).left ⟶ ((Over.pullback f).obj Y).left)
⊢ pullback.map X.hom f Y.hom f (Over.Hom.left b) (𝟙 T) (𝟙 S) ⋯ ⋯ ≫ pullback.snd Y.hom f =
    Over.Hom.left ((Over.pullback f).map b) ≫ pullback.snd Y.hom f

Tactic:
<;>

Goal state after:
case e'_2.h₀
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
P : MorphismProperty C
inst✝³ : (isomorphisms C).DescendsAlong P
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : HasPullbacks C
inst✝ : HasEqualizers C
S T : C
f : T ⟶ S
hf : P f
X Y : Over S
a b : X ⟶ Y
hab : (Over.pullback f).map a = (Over.pullback f).map b
e_1✝ : (pullback X.hom f ⟶ pullback Y.hom f) = (((Over.pullback f).obj X).left ⟶ ((Over.pullback f).obj Y).left)
⊢ pullback.map X.hom f Y.hom f (Over.Hom.left a) (𝟙 T) (𝟙 S) ⋯ ⋯ ≫ pullback.fst Y.hom f =
    Over.Hom.left ((Over.pullback f).map a) ≫ pullback.fst Y.hom f

case h₁
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
P : MorphismProperty C
inst✝³ : (isomorphisms C).DescendsAlong P
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : HasPullbacks C
inst✝ : HasEqualizers C
S T : C
f : T ⟶ S
hf : P f
X Y : Over S
a b : X ⟶ Y
hab : (Over.pullback f).map a = (Over.pullback f).map b
e_1✝ : (pullback X.hom f ⟶ pullback Y.hom f) = (((Over.pullback f).obj X).left ⟶ ((Over.pullback f).obj Y).left)
⊢ pullback.map X.hom f Y.hom f (Over.Hom.left a) (𝟙 T) (𝟙 S) ⋯ ⋯ ≫ pullback.snd Y.hom f =
    Over.Hom.left ((Over.pullback f).map a) ≫ pullback.snd Y.hom f

case e'_3.h₀
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
P : MorphismProperty C
inst✝³ : (isomorphisms C).DescendsAlong P
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : HasPullbacks C
inst✝ : HasEqualizers C
S T : C
f : T ⟶ S
hf : P f
X Y : Over S
a b : X ⟶ Y
hab : (Over.pullback f).map a = (Over.pullback f).map b
e_1✝ : (pullback X.hom f ⟶ pullback Y.hom f) = (((Over.pullback f).obj X).left ⟶ ((Over.pullback f).obj Y).left)
⊢ pullback.map X.hom f Y.hom f (Over.Hom.left b) (𝟙 T) (𝟙 S) ⋯ ⋯ ≫ pullback.fst Y.hom f =
    Over.Hom.left ((Over.pullback f).map b) ≫ pullback.fst Y.hom f

case h₁
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
P : MorphismProperty C
inst✝³ : (isomorphisms C).DescendsAlong P
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : HasPullbacks C
inst✝ : HasEqualizers C
S T : C
f : T ⟶ S
hf : P f
X Y : Over S
a b : X ⟶ Y
hab : (Over.pullback f).map a = (Over.pullback f).map b
e_1✝ : (pullback X.hom f ⟶ pullback Y.hom f) = (((Over.pullback f).obj X).left ⟶ ((Over.pullback f).obj Y).left)
⊢ pullback.map X.hom f Y.hom f (Over.Hom.left b) (𝟙 T) (𝟙 S) ⋯ ⋯ ≫ pullback.snd Y.hom f =
    Over.Hom.left ((Over.pullback f).map b) ≫ pullback.snd Y.hom f