English
Composition of two homCongr operations corresponds to a two-step congruence decomposition.
Русский
Сложение двух операций homCongr эквивалентно двухступенчатому разложению конгруэнций.
LaTeX
$$$ (\\alpha_1 \\ll\\gg \\alpha_2).homCongr (\\beta_1 \\ll\\gg \\beta_2) f = (\\alpha_1.homCongr \\beta_1).trans (\\alpha_2.homCongr \\beta_2) f $$$
Lean4
/-- If `X` is isomorphic to `X₁` and `Y` is isomorphic to `Y₁`, then
there is a bijection between `X ≅ Y` and `X₁ ≅ Y₁`. -/
@[simps]
def isoCongr {X₁ Y₁ X₂ Y₂ : C} (f : X₁ ≅ X₂) (g : Y₁ ≅ Y₂) : (X₁ ≅ Y₁) ≃ (X₂ ≅ Y₂)
where
toFun h := f.symm.trans <| h.trans <| g
invFun h := f.trans <| h.trans <| g.symm
left_inv := by cat_disch
right_inv := by cat_disch
