  The classical double copy relating exact solutions of biadjoint scalar, gauge
and gravity theories continues to receive widespread attention. Recently, a
derivation of the exact classical double copy was presented, using ideas from
twistor theory, in which spacetime fields are mapped to Cech cohomology classes
in twistor space. A puzzle remains, however, in how to interpret the twistor
double copy, in that it relies on somehow picking special representatives of
each cohomology class. In this paper, we provide two alternative formulations
of the twistor double copy using the more widely-used language of Dolbeault
cohomology. The first amounts to a rewriting of the Cech approach, whereas the
second uses known techniques for discussing spacetime fields in Euclidean
signature. The latter approach indeed allows us to identify special cohomology
representatives, suggesting that further application of twistor methods in
exploring the remit of the double copy may be fruitful.
