When the class
σ
\sigma
of bi-univalent functions was first defined, it was known that functions of the form
ϕ
∘
ψ
−
1
∈
σ
\phi \circ {\psi ^{ - 1}} \in \sigma
when
ϕ
\phi
and
ψ
\psi
are univalent, map the unit disc
B
{\mathbf {B}}
onto a set containing
B
{\mathbf {B}}
, and satisfy
ϕ
(
0
)
=
ψ
(
0
)
=
0
\phi (0) = \psi (0) = 0
,
ϕ
′
(
0
)
=
ψ
′
(
0
)
\phi ’(0) = \psi ’(0)
. It is shown here that such functions form a proper subset of
σ
\sigma
, and that
σ
\sigma
is a proper subset of the set of functions of the form
ϕ
∘
ψ
−
1
\phi \circ {\psi ^{ - 1}}
, where
ϕ
\phi
and
ψ
\psi
are locally univalent, at most
2
2
-valent, each maps a subregion of
B
{\mathbf {B}}
univalently onto
B
{\mathbf {B}}
, and
ϕ
(
0
)
=
ψ
(
0
)
=
0
\phi (0) = \psi (0) = 0
,
ϕ
′
(
0
)
=
ψ
′
(
0
)
\phi ’(0) = \psi ’(0)
,
ψ
−
1
(
0
)
=
0
{\psi ^{ - 1}}(0) = 0
. It is also shown that there are
f
(
z
)
=
z
+
a
2
z
2
+
⋯
f(z) = z + {a_2}{z^2} + \cdots
in
σ
\sigma
with
|
a
2
|
>
4
/
3
\left | {{a_2}} \right | > 4/3
. However, doubt is cast that
|
a
2
|
\left | {{a_2}} \right |
can be as large as
3
/
2
3/2
.