  Okounkov and Pandharipande proved that the equivariant Toda hierarchy governs
the equivariant Gromov-Witten theory of $\mathbb{CP}^1$. A technical clue of
their method is a pair of dressing operators on the Fock space of 2D charged
free fermion fields. We reformulate these operators as difference operators in
the Lax formalism of the 2D Toda hierarchy. This leads to a new explanation to
the question of why the equivariant Toda hierarchy emerges in the equivariant
Gromov-Witten theory of $\mathbb{CP}^1$. Moreover, the non-equivariant limit of
these operators turns out to capture the integrable structure of the
non-equivariant Gromov-Witten theory correctly.
