English
essImage maps monoid objects equivalently to essImage on underlying objects under fully faithful functors.
Русский
Эс Image сохраняет отображение моноидных объектов эквивализованно к эс Image базовых объектов для полностью верного функторa.
LaTeX
$$$F.mapMon.essImage M \\leftrightarrow F.essImage M.X$$$
Lean4
/-- The essential image of a fully faithful functor between cartesian-monoidal categories is the
same on monoid objects as on objects. -/
@[simp]
theorem essImage_mapMon [F.Full] [F.Faithful] {M : Mon D} : F.mapMon.essImage M ↔ F.essImage M.X
where
mp := by rintro ⟨N, ⟨e⟩⟩; exact ⟨N.X, ⟨(Mon.forget _).mapIso e⟩⟩
mpr := by
rintro ⟨N, ⟨e⟩⟩
let : MonObj (F.obj N) := .ofIso e.symm
let : MonObj N := (FullyFaithful.ofFullyFaithful F).monObj N
refine ⟨.mk N, ⟨Mon.mkIso e ?_ ?_⟩⟩ <;> simp
