  Let $f$ be a non-invertible partially hyperbolic endomorphism on
$\mathbb{T}^2$ which is derived from a non-expanding Anosov endomorphism.
Differing from the case of diffeomorphisms derived from Anosov automorphisms,
there is no a priori semi-conjugacy between $f$ and its linearization on
$\mathbb{T}^2$. We show that $f$ is semi-conjugate to its linearization if and
only if $f$ admits the partially hyperbolic splitting with two $Df$-invariant
subbundles.
  Moreover, if we assume that $f$ has the unstable subbundle, then the
semi-conjugacy is exactly a topological conjugacy, and the center Lyapunov
exponents of periodic points of $f$ coincide and equal to the stable Lyapunov
exponent of its linearization. In particular, $f$ is an Anosov endomorphism and
the conjugacy is smooth along the stable foliation.
  For the case that $f$ has the stable subbundle, there is still some rigidity
on its stable Lyapunov exponents. However, we also give examples which admit
the partially hyperbolic splitting with the center subbundle but the
semi-conjugacy is indeed non-injective.
  Finally, We present some applications for the conservative case. In
particular, we show that if $f$ is volume-preserving and semi-conjugate to its
linear part, then the semi-conjugacy is actually a smooth conjugacy.
