  The log-partition function $ \log W_N(\beta)$ of the two-dimensional directed
polymer in random environment is known to converge in distribution to a normal
distribution when considering temperature in the subcritical regime
$\beta=\beta_N=\hat{\beta}\sqrt{\pi/\log N}$, $\hat{\beta}\in (0,1)$
(Caravenna, Sun, Zygouras, Ann. Appl. Prob. (2017)). In this paper, we present
an elementary proof of this result relying on a decoupling argument and the
central limit theorem for sums of independent random variables. The argument is
inspired by an analogy of the model to branching random walks.
