/-- Adding a first point is left adjoint to forgetting the first point. -/
noncomputable def pointedToTwoPSndForgetCompBipointedToPointedSndAdjunction :
    pointedToTwoPSnd ⊣ forget₂ TwoP Bipointed ⋙ bipointedToPointedSnd :=
  Adjunction.mkOfHomEquiv
    { homEquiv := fun X Y =>
        { toFun := fun f => ⟨f.toFun ∘ Option.some, f.map_snd⟩
          invFun := fun f => ⟨fun o => o.elim Y.toTwoPointing.toProd.1 f.toFun, rfl, f.map_point⟩
          left_inv := fun f => by
            apply Bipointed.Hom.ext
            funext x
            cases x
            exact f.map_fst.symm
            rfl
          right_inv := fun f => Pointed.Hom.ext _ _ rfl }
      homEquiv_naturality_left_symm := fun f g => by
        apply Bipointed.Hom.ext
        funext x
        cases x <;> 

Proof state:
case toFun.h.none
α : Type u_1
β : Type u_2
X'✝ X✝ : Pointed
Y✝ : TwoP
f : X'✝ ⟶ X✝
g : X✝ ⟶ (forget₂ TwoP Bipointed ⋙ bipointedToPointedSnd).obj Y✝
⊢ Bipointed.Hom.toFun
      (((fun X Y =>
              {
                toFun := fun f =>
                  { toFun := f.toFun ∘ some,
                    map_point :=
                      (_ :
                        Bipointed.Hom.toFun f (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.2 =
                          (TwoP.toBipointed Y).toProd.2) },
                invFun := fun f =>
                  { toFun := fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun,
                    map_fst :=
                      (_ :
                        (fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun)
                            (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.1 =
                          (fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun)
                            (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.1),
                    map_snd :=
                      (_ :
                        Pointed.Hom.toFun f X.point = ((forget₂ TwoP Bipointed ⋙ bipointedToPointedSnd).obj Y).point) },
                left_inv :=
                  (_ :
                    ∀ (f : pointedToTwoPSnd.obj X ⟶ Y),
                      (fun f =>
                            { toFun := fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun,
                              map_fst :=
                                (_ :
                                  (fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun)
                                      (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.1 =
                                    (fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun)
                                      (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.1),
                              map_snd :=
                                (_ :
                                  Pointed.Hom.toFun f X.point =
                                    ((forget₂ TwoP Bipointed ⋙ bipointedToPointedSnd).obj Y).point) })
                          ((fun f =>
                              { toFun := f.toFun ∘ some,
                                map_point :=
                                  (_ :
                                    Bipointed.Hom.toFun f (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.2 =
                                      (TwoP.toBipointed Y).toProd.2) })
                            f) =
                        f),
                right_inv :=
                  (_ :
                    ∀ (f : X ⟶ (forget₂ TwoP Bipointed ⋙ bipointedToPointedSnd).obj Y),
                      (fun f =>
                            { toFun := f.toFun ∘ some,
                              map_point :=
                                (_ :
                                  Bipointed.Hom.toFun f (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.2 =
                                    (TwoP.toBipointed Y).toProd.2) })
                          ((fun f =>
                              { toFun := fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun,
                                map_fst :=
                                  (_ :
                                    (fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun)
                                        (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.1 =
                                      (fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun)
                                        (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.1),
                                map_snd :=
                                  (_ :
                                    Pointed.Hom.toFun f X.point =
                                      ((forget₂ TwoP Bipointed ⋙ bipointedToPointedSnd).obj Y).point) })
                            f) =
                        f) })
            X'✝ Y✝).symm
        (f ≫ g))
      none =
    Bipointed.Hom.toFun
      (pointedToTwoPSnd.map f ≫
        ((fun X Y =>
                {
                  toFun := fun f =>
                    { toFun := f.toFun ∘ some,
                      map_point :=
                        (_ :
                          Bipointed.Hom.toFun f (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.2 =
                            (TwoP.toBipointed Y).toProd.2) },
                  invFun := fun f =>
                    { toFun := fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun,
                      map_fst :=
                        (_ :
                          (fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun)
                              (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.1 =
                            (fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun)
                              (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.1),
                      map_snd :=
                        (_ :
                          Pointed.Hom.toFun f X.point =
                            ((forget₂ TwoP Bipointed ⋙ bipointedToPointedSnd).obj Y).point) },
                  left_inv :=
                    (_ :
                      ∀ (f : pointedToTwoPSnd.obj X ⟶ Y),
                        (fun f =>
                              { toFun := fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun,
                                map_fst :=
                                  (_ :
                                    (fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun)
                                        (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.1 =
                                      (fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun)
                                        (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.1),
                                map_snd :=
                                  (_ :
                                    Pointed.Hom.toFun f X.point =
                                      ((forget₂ TwoP Bipointed ⋙ bipointedToPointedSnd).obj Y).point) })
                            ((fun f =>
                                { toFun := f.toFun ∘ some,
                                  map_point :=
                                    (_ :
                                      Bipointed.Hom.toFun f (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.2 =
                                        (TwoP.toBipointed Y).toProd.2) })
                              f) =
                          f),
                  right_inv :=
                    (_ :
                      ∀ (f : X ⟶ (forget₂ TwoP Bipointed ⋙ bipointedToPointedSnd).obj Y),
                        (fun f =>
                              { toFun := f.toFun ∘ some,
                                map_point :=
                                  (_ :
                                    Bipointed.Hom.toFun f (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.2 =
                                      (TwoP.toBipointed Y).toProd.2) })
                            ((fun f =>
                                { toFun := fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun,
                                  map_fst :=
                                    (_ :
                                      (fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun)
                                          (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.1 =
                                        (fun o => Option.elim o Y.toTwoPointing.toProd.1 f.toFun)
                                          (TwoP.toBipointed (pointedToTwoPSnd.obj X)).toProd.1),
                                  map_snd :=
                                    (_ :
                                      Pointed.Hom.toFun f X.point =
                                        ((forget₂ TwoP Bipointed ⋙ bipointedToPointedSnd).obj Y).point) })
                              f) =
                          f) })
              X✝ Y✝).symm
          g)
      none

Next tactic:
rfl