English
The counit of the adjunction between condensed sets and topological spaces has a canonical evaluation property on PUnit: for X in TopCat and x ∈ C(PUnit, X), the underlying map sends x to its value at PUnit.unit.
Русский
Существaет каноническое свойство оценки для компоненту counit при PUnit: для X в TopCat и x ∈ C(PUnit, X) отображение выводит x в значение на PUnit.unit.
LaTeX
$$$\text{topCatAdjunctionCounit}_X(x) = x(PUnit.unit)$$$
Lean4
/-- `simp`-normal form of the lemma that `@[simps]` would generate. -/
@[simp]
theorem topCatAdjunctionCounit_hom_apply (X : TopCat) (x) :
-- We have to specify here to not infer the `TopologicalSpace` instance on `C(PUnit, X)`,
-- which suggests type synonyms are being unfolded too far somewhere.
DFunLike.coe (F := @ContinuousMap C(PUnit, X) X (_) _) (TopCat.Hom.hom (topCatAdjunctionCounit X)) x =
x PUnit.unit :=
rfl
