We consider moduli spaces of dynamical systems of correspondences over the projective line as a generalization of moduli spaces of dynamical systems of endomorphisms on the projective line. We define the moduli space
D
y
n
d
,
e
Dyn_{d,e}
of degree
(
d
,
e
)
(d,e)
correspondences. We construct a family
ρ
c
:
D
y
n
d
,
e
⇢
D
y
n
1
,
d
+
e
−
1
\rho _c : Dyn_{d,e} \dashrightarrow Dyn_{1,d+e-1}
of rational maps representation-theoretically. Here we note that
D
y
n
1
,
d
+
e
−
1
Dyn_{1,d+e-1}
is identical to the moduli space of the usual dynamical systems of degree
d
+
e
−
1
d+e-1
. We show that the moduli space
D
y
n
d
,
e
Dyn_{d,e}
is rational by using
ρ
c
\rho _c
. Moreover, the multiplier maps for the fixed points factor through
ρ
c
\rho _c
. Furthermore, we show the Woods Hole formulae for correspondences of different degrees are also related by
ρ
c
\rho _c
and obtain another representation-theoretically simplified form of the formula.