= Three-point amputation in the connected-action convention
{title2=$G_3^c=i\Gamma^{(3)}(G_2^c)^{\otimes3}$}
Use the Minkowski <generating functional> $Z=e^{iW}$, source $+J\phi$, mean field $\varphi=\delta W/\delta J$, and <quantum effective action> $\Gamma=W-\int J\varphi$. Then $W^{(2)}=iG_2^c$, $W^{(3)}=-G_3^c$, and $\Gamma^{(2)}=-(W^{(2)})^{-1}$. Differentiating this inverse gives $\Gamma^{(3)}=(W^{(2)})^{-1\otimes3}W^{(3)}=-i(G_2^c)^{-1\otimes3}G_3^c$, with integration over the repeated spacetime arguments. Thus <amputation of external propagators> gives $i\Gamma^{(3)}$ in this convention. The full connected two-point functions, rather than bare propagators, perform the exact amputation.
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