  Both the Bern, Carrasco and Johansson (BCJ) and the Kawai, Lewellen and Tye
(KLT) double-copy formalisms have been recently generalized to a class of
scattering matrix elements (so-called form factors) that involve local
gauge-invariant operators. In this paper we continue the study of double copy
for form factors. First, we generalize the double-copy prescription to form
factors of higher-length operators ${\rm tr}(\phi^m)$ with $m\geq3$. These
higher-length operators introduce new non-trivial color identities, but the
double-copy prescription works perfectly well. The closed formulae for the
CK-dual numerators are also provided. Next, we discuss the $\vec{v}$ vectors
which are central ingredients appearing in the factorization relations of both
the KLT kernels and the gauge form factors. We present a general construction
rule for the $\vec{v}$ vectors and discuss their universal properties. Finally,
we consider the double copy for the form factor of the ${\rm tr}(F^2)$ operator
in pure Yang-Mills theory. In this case, we propose a new prescription that
involves a gauge invariant decomposition for the form factor and a combination
of different CK-dual numerators appearing in the expansion.
