  This article deals with the tamely ramified geometric Langlands
correspondence for GL_2 on $\mathbf{P}_{\mathbf{F}_q}^1$, where $q$ is a prime
power, with tame ramification at four distinct points $D = \{\infty, 0,1, t\}
\subset \mathbf{P}^1(\mathbf{F}_q)$. We describe in an explicit way (1) the
action of the Hecke operators on a basis of the cusp forms, which consists of
$q$ elements; and (2) the correspondence that assigns to a pure irreducible
rank 2 local system $E$ on $\mathbf{P}^1 \setminus D$ with unipotent monodromy
its Hecke eigensheaf on the moduli space of rank 2 parabolic vector bundles. We
define a canonical embedding $\mathbf{P}^1$ into this module space and show
with a new proof that the restriction of the eigensheaf to the degree 1 part of
this moduli space is the intermediate extension of $E$.
