= Solution
Applying four functional derivatives to the functional equation gives the homogeneous equation for the renormalized <four-point one-particle-irreducible correlation function>:
$$
\boxed{\left[
\Lambda\partial_\Lambda+\beta_{m^2}\partial_{m^2}
+\sum_i\beta_i\partial_{g_i}-4\gamma_\phi
\right]
\Gamma_\Lambda^{(4)}(x_1,x_2,x_3,x_4;m^2,g_i)=0.}
$$
If the opposite convention for $\gamma_\phi$ is used, the last sign changes. The factor four counts the four external renormalized fields.
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