  We introduce the notion of directed hereditary species and show that they
have associated monoidal decomposition spaces, comodule bialgebras, and
operadic categories. The notion subsumes Schmitt's hereditary species,
G\'alvez--Kock--Tonks directed restrictions species, and a directed version of
Carlier's construction of monoidal decomposition spaces and comodule
bialgebras. In addition to all the examples of Schmitt, G\'alvez--Kock--Tonks
and Carlier, the new construction covers also the Fauvet--Foissy--Manchon
comodule bialgebra of finite topological spaces, the
Calaque--Ebrahimi-Fard--Manchon comodule bialgebra of rooted trees, and the
Fa\`a di Bruno comodule bialgebra of linear trees.
