  We consider the component structure of the random digraph $D(n,p)$ inside the
critical window $p = n^{-1} + \lambda n^{-4/3}$.We show that the largest
component $\mathcal{C}_1$ has size of order $n^{1/3}$ in this range. In
particular we give explicit bounds on the tail probabilities of
$|\mathcal{C}_1|n^{-1/3}$.
