/-- `typeToPointed` is the free functor. -/
def typeToPointedForgetAdjunction : typeToPointed ⊣ forget Pointed :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun X Y =>
        { toFun := fun f => f.toFun ∘ Option.some
          invFun := fun f => ⟨fun o => o.elim Y.point f, rfl⟩
          left_inv := fun f => by
            apply Pointed.Hom.ext
            funext x
            cases x
            · exact f.map_point.symm
            · rfl
          right_inv := fun f => funext fun _ => rfl }
      homEquiv_naturality_left_symm := fun f g => by
        apply Pointed.Hom.ext
        

Proof state:
case toFun
α : Type u_1
β : Type u_2
X'✝ X✝ : Type ?u.7456
Y✝ : Pointed
f : X'✝ ⟶ X✝
g : X✝ ⟶ (forget Pointed).obj Y✝
⊢ (((fun X Y =>
              { toFun := fun f => f.toFun ∘ some,
                invFun := fun f =>
                  { toFun := fun o => Option.elim o Y.point f,
                    map_point :=
                      (_ :
                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                          (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) },
                left_inv :=
                  (_ :
                    ∀ (f : typeToPointed.obj X ⟶ Y),
                      (fun f =>
                            { toFun := fun o => Option.elim o Y.point f,
                              map_point :=
                                (_ :
                                  (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                    (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                          ((fun f => f.toFun ∘ some) f) =
                        f),
                right_inv :=
                  (_ :
                    ∀ (f : X ⟶ (forget Pointed).obj Y),
                      (fun f => f.toFun ∘ some)
                          ((fun f =>
                              { toFun := fun o => Option.elim o Y.point f,
                                map_point :=
                                  (_ :
                                    (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                            f) =
                        f) })
            X'✝ Y✝).symm
        (f ≫ g)).toFun =
    (typeToPointed.map f ≫
        ((fun X Y =>
                { toFun := fun f => f.toFun ∘ some,
                  invFun := fun f =>
                    { toFun := fun o => Option.elim o Y.point f,
                      map_point :=
                        (_ :
                          (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                            (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) },
                  left_inv :=
                    (_ :
                      ∀ (f : typeToPointed.obj X ⟶ Y),
                        (fun f =>
                              { toFun := fun o => Option.elim o Y.point f,
                                map_point :=
                                  (_ :
                                    (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                            ((fun f => f.toFun ∘ some) f) =
                          f),
                  right_inv :=
                    (_ :
                      ∀ (f : X ⟶ (forget Pointed).obj Y),
                        (fun f => f.toFun ∘ some)
                            ((fun f =>
                                { toFun := fun o => Option.elim o Y.point f,
                                  map_point :=
                                    (_ :
                                      (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point =
                                        (fun o => Option.elim o Y.point f) (typeToPointed.obj X).point) })
                              f) =
                          f) })
              X✝ Y✝).symm
          g).toFun

Next tactic:
funext x