English
The extensionality lemma for coind' maps ensures equality of coind' morphisms from equality of their action on generators.
Русский
Лемма экстенсиональности для отображений коинд' гарантирует равенство коинд' гомоморфизмов по равенству их действия на генераторы.
LaTeX
$$$\text{coind'}\text{ext} : \ f=g \text{ iff } f(1)=g(1)$$$
Lean4
/-- Given a monoid morphism `φ : G →* H` and a morphism of `G`-representations `f : A ⟶ B`, there
is a natural `H`-representation morphism `coind' φ A ⟶ coind' φ B`, given by postcomposition
by `f`. -/
@[simps]
noncomputable def coindMap' {A B : Rep k G} (f : A ⟶ B) : coind' φ A ⟶ coind' φ B
where
hom := ModuleCat.ofHom <| Linear.rightComp k _ f
comm h := by ext; simp [ModuleCat.endRingEquiv]
