  We consider the strongly connected components (SCCs) of a uniform directed
graph on $n$ vertices with i.i.d. in- and out-degree pairs distributed as
$(D^-,D^+)$, with $\mathbb E[D^+]=\mathbb E[D^-]=\mu$. We condition on equal
total in- and out-degree. A phase transition for the emergence of a giant SCC
is known to occur at the critical value $\mathbb E[D^-D^+] = \mu$. We study the
model at this critical value and, additionally, require that $\mathbb
E[(D^-)^i(D^+)^j]<\infty$ for all $i+j\leq 3$, and for $(i,j)=(1,3)$ and
$(i,j)=(3,1)$. We show that, under these conditions, the SCCs ranked by
decreasing number of edges with distances rescaled by $n^{-1/3}$ converge in
distribution to a sequence of finite strongly connected directed multigraphs
with edge lengths, and that these are either $3$-regular or loops. The limit
objects lie in a $3$-parameter family, which contains the scaling limit of the
SCCs in the directed Erd\H{o}s-R\'enyi model at criticality as found by
Goldschmidt and Stephenson (2019). This is the first universality result for
the scaling limit of a critical directed graph model and the first quantitative
result on the directed configuration model at criticality. As a trivial
consequence, the largest SCCs at criticality contain $\Theta(n^{1/3})$ vertices
and edges in probability, and the diameter of the directed graph at criticality
is $\Omega(n^{1/3})$ in probability. We use a metric on the space of weighted
multigraphs in which two multigraphs are close if there are compatible
isomorphisms between their vertex and edge sets which roughly preserve the edge
lengths. We use the product topology on the sequence of multigraphs. Our method
of proof involves a depth-first exploration of the directed graph, resulting in
a spanning forest with additional identifications, of which we study the limit
under rescaling.
