  In this work, we introduce explicit formulae for the coproducts at play for
multi-indices in ODEs and in singular SPDEs. The two coproducts described
correspond to versions of the Butcher-Connes-Kreimer and extraction/contraction
coproducts with multi-indices. The main idea is to use the fact that these
coproducts are the adjoints of dual products for which one has explicit simple
formulae. We are able to derive the explicit formulae via an inner product
defined from a symmetry factor easily computable for multi-indices.
