symmetry — Mathlib

English

Swapping twice is naturally isomorphic to the identity on C × D: swap ∘ swap ≅ Id_{C×D}.

Русский

Поменять два раза равно природно гомоморфизируемо к тождественному на C × D: swap ∘ swap ≅ Id_{C×D}.

LaTeX

$$$\mathrm{swap}_{C,D} \circ \mathrm{swap}_{D,C} \cong \mathrm{Id}_{C\times D}$$$

Lean4

/-- Swapping the factors of a Cartesian product of categories twice is naturally isomorphic
to the identity functor.
-/
@[simps]
def symmetry : swap C D ⋙ swap D C ≅ 𝟭 (C × D)
    where
  hom := { app := fun X => 𝟙 X }
  inv := { app := fun X => 𝟙 X }

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