  Aguiar and Mahajan introduced a cohomology theory for the twisted coalgebras
of Joyal, with particular interest in the computation of their second
cohomology group, which gives rise to their deformations. We use the Koszul
duality theory between twisted algebras and coalgebras on the twisted coalgebra
that gives rise to their cohomology theory to give a new alternative
description of it which, in particular, allows for its effective computation.
We compute it completely in various examples, including those proposed by
Aguiar and Mahajan, and obtain structural results: in particular, we study its
multiplicative structure and provide a K\"unneth formula.
