  For any coloured operad R, we prove a Fa\`a di Bruno formula for the
`connected Green function' in the incidence bialgebra of R. This generalises on
one hand the classical Fa\`a di Bruno formula (dual to composition of power
series), corresponding to the case where R is the terminal reduced operad, and
on the other hand the Faa di Bruno formula for P-trees of G\'alvez--Kock--Tonks
(P a finitary polynomial endofunctor), which corresponds to the case where R is
the free operad on P. Following G\'alvez--Kock--Tonks, we work at the objective
level of groupoid slices, hence all proofs are `bijective': the formula is
established as the homotopy cardinality of an explicit equivalence of
groupoids. In fact we establish the formula more generally in a relative
situation, for algebras for one polynomial monad internal to another. This
covers in particular nonsymmetric operads (for which the terminal reduced case
yields the noncommutative Fa\`a di Bruno formula of
Brouder--Frabetti--Krattenthaler).
