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We propose a general coaction for families of integrals appearing in theevaluation of Feynman diagrams, such as multiple polylogarithms and generalizedhypergeometric functions. We further conjecture a link between this coactionand graphical operations on Feynman diagrams. At one-loop order, there is abasis of integrals for which this correspondence is fully explicit. We discussfeatures and present examples of the diagrammatic coaction on two-loopintegrals. We also present the coaction for the functions ${}_{p+1}F_p$ andAppell $F_1$.

Coaction for Feynman integrals and diagrams