  We give the crystal structure of the Grothendieck group $G_0(R)$ of
irreducible modules over the quiver Hecke algebra $R$ constructed in
\cite{TW2023}. This leads to the categorification of the crystal $B(\infty)$ of
the quantum Borcherds algebra $U_q(\mathscr g)$ and its irreducible highest
weight crystal $B(\lambda)$ for arbitrary Borcherds-Cartan data. Additionally,
we study the cyclotomic categorification of irreducible highest weight
$U_q(\mathscr g)$-modules.
