instance : FunLike (A →ₙₐ[R] B) A fun _ => B where
  coe f := f.toFun
  coe_injective' := by rintro ⟨⟨⟨f, _⟩, _⟩, _⟩ ⟨⟨⟨g, _⟩, _⟩, _⟩ h; 

Proof state:
case mk.mk.mk.mk.mk.mk
R : Type u
A : Type v
B : Type w
C : Type w₁
inst✝⁶ : Monoid R
inst✝⁵ : NonUnitalNonAssocSemiring A
inst✝⁴ : DistribMulAction R A
inst✝³ : NonUnitalNonAssocSemiring B
inst✝² : DistribMulAction R B
inst✝¹ : NonUnitalNonAssocSemiring C
inst✝ : DistribMulAction R C
f : A → B
map_smul'✝¹ : ∀ (m : R) (x : A), f (m • x) = m • f x
map_zero'✝¹ : MulActionHom.toFun { toFun := f, map_smul' := map_smul'✝¹ } 0 = 0
map_add'✝¹ :
  ∀ (x y : A),
    MulActionHom.toFun { toFun := f, map_smul' := map_smul'✝¹ } (x + y) =
      MulActionHom.toFun { toFun := f, map_smul' := map_smul'✝¹ } x +
        MulActionHom.toFun { toFun := f, map_smul' := map_smul'✝¹ } y
map_mul'✝¹ :
  ∀ (x y : A),
    MulActionHom.toFun
        { toMulActionHom := { toFun := f, map_smul' := map_smul'✝¹ }, map_zero' := map_zero'✝¹,
            map_add' := map_add'✝¹ }.toMulActionHom
        (x * y) =
      MulActionHom.toFun
          { toMulActionHom := { toFun := f, map_smul' := map_smul'✝¹ }, map_zero' := map_zero'✝¹,
              map_add' := map_add'✝¹ }.toMulActionHom
          x *
        MulActionHom.toFun
          { toMulActionHom := { toFun := f, map_smul' := map_smul'✝¹ }, map_zero' := map_zero'✝¹,
              map_add' := map_add'✝¹ }.toMulActionHom
          y
g : A → B
map_smul'✝ : ∀ (m : R) (x : A), g (m • x) = m • g x
map_zero'✝ : MulActionHom.toFun { toFun := g, map_smul' := map_smul'✝ } 0 = 0
map_add'✝ :
  ∀ (x y : A),
    MulActionHom.toFun { toFun := g, map_smul' := map_smul'✝ } (x + y) =
      MulActionHom.toFun { toFun := g, map_smul' := map_smul'✝ } x +
        MulActionHom.toFun { toFun := g, map_smul' := map_smul'✝ } y
map_mul'✝ :
  ∀ (x y : A),
    MulActionHom.toFun
        { toMulActionHom := { toFun := g, map_smul' := map_smul'✝ }, map_zero' := map_zero'✝,
            map_add' := map_add'✝ }.toMulActionHom
        (x * y) =
      MulActionHom.toFun
          { toMulActionHom := { toFun := g, map_smul' := map_smul'✝ }, map_zero' := map_zero'✝,
              map_add' := map_add'✝ }.toMulActionHom
          x *
        MulActionHom.toFun
          { toMulActionHom := { toFun := g, map_smul' := map_smul'✝ }, map_zero' := map_zero'✝,
              map_add' := map_add'✝ }.toMulActionHom
          y
h :
  (fun f => f.toFun)
      {
        toDistribMulActionHom :=
          { toMulActionHom := { toFun := f, map_smul' := map_smul'✝¹ }, map_zero' := map_zero'✝¹,
            map_add' := map_add'✝¹ },
        map_mul' := map_mul'✝¹ } =
    (fun f => f.toFun)
      {
        toDistribMulActionHom :=
          { toMulActionHom := { toFun := g, map_smul' := map_smul'✝ }, map_zero' := map_zero'✝, map_add' := map_add'✝ },
        map_mul' := map_mul'✝ }
⊢ {
      toDistribMulActionHom :=
        { toMulActionHom := { toFun := f, map_smul' := map_smul'✝¹ }, map_zero' := map_zero'✝¹,
          map_add' := map_add'✝¹ },
      map_mul' := map_mul'✝¹ } =
    {
      toDistribMulActionHom :=
        { toMulActionHom := { toFun := g, map_smul' := map_smul'✝ }, map_zero' := map_zero'✝, map_add' := map_add'✝ },
      map_mul' := map_mul'✝ }

Next tactic:
congr