case fac
C : Type u_2
inst✝² : Category C
inst✝¹ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
⊢ factorThruImage (θ.unop ≫ Splitting.IndexSet.e A) ≫ image.ι (θ.unop ≫ Splitting.IndexSet.e A) =
    θ.unop ≫ Splitting.IndexSet.e A
C : Type u_2
inst✝² : Category C
inst✝¹ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
⊢ Splitting.ιSummand (splitting K) A ≫ (obj K).map θ =
    Termwise.mapMono K (image.ι (θ.unop ≫ Splitting.IndexSet.e A)) ≫
      Splitting.ιSummand (splitting K) (Splitting.IndexSet.pull A θ)
apply Obj.map_on_summand
no goals
case fac
C : Type u_2
inst✝² : Category C
inst✝¹ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ'' : SimplexCategory
inst✝ : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
⊢ factorThruImage (θ.unop ≫ Splitting.IndexSet.e A) ≫ image.ι (θ.unop ≫ Splitting.IndexSet.e A) =
    θ.unop ≫ Splitting.IndexSet.e A
apply image.fac