English
In a closed monoidal category, the transpose composition satisfies a concrete formula: compTranspose x y z equals the inverse associator composed with ihom.ev x and ihom.ev y applications.
Русский
В замкнутой монааловой категории транспонированное сопоставление удовлетворяет конкретной формуле: compTranspose x y z равно обратному ассоциатору, сочетаемому с ihom.ev x и ihom.ev y.
LaTeX
$$$ \\mathrm{compTranspose}(x,y,z) = (\\alpha_{x,y,z})^{-1} \\circ (\\mathrm{ihom.ev}\ x)_{y} \\triangleright (\\mathrm{ihom.ev}\ y)_{z} $$$
Lean4
theorem mk_eq_mk_iff {X Y X' Y' : T} (f : X ⟶ Y) (f' : X' ⟶ Y') :
Arrow.mk f = Arrow.mk f' ↔ ∃ (hX : X = X') (hY : Y = Y'), f = eqToHom hX ≫ f' ≫ eqToHom hY.symm :=
by
constructor
· intro h
refine ⟨congr_arg Comma.left h, congr_arg Comma.right h, ?_⟩
have := (eqToIso h).hom.w
dsimp at this
rw [Comma.eqToHom_left, Comma.eqToHom_right] at this
rw [reassoc_of% this, eqToHom_trans, eqToHom_refl, Category.comp_id]
· rintro ⟨rfl, rfl, h⟩
simp only [eqToHom_refl, Category.comp_id, Category.id_comp] at h
rw [h]
