  We summarise the effective field theory of dark energy construction to
explore observable predictions of linear Horndeski theories. Based on
\cite{Perenon:2016blf}, we review the diagnostic of these theories on the
correlation of the large-scale structure phenomenological functions: the
effective Newton constant, the light deflection parameter and the growth
function of matter perturbations. We take this opportunity to discuss the
evolution of the bounds the propagation speed of gravitational waves has
undergone and use the most restrictive one to update the diagnostic.
