What Ising coupling does the $\phi^4$ coupling correspond to?

It is well known that the $\phi^4$ model and the Ising model in 2 dimensions are in the same universality class. The Ising model at the critical point has something like $\sum_{}\sigma_i \sigma_j$ coupled to something like $(T-T_c)$, or, alternatively, if we use the quantum TFIM at 0 temperature, something like this
$
\mathcal{L} = -J \sum_{} Z_i Z_j - h \sum_{i}X_i
$which reaches the critical point when $J=h$.

The $\phi^4-$model has 2 parameters, the mass and the coupling $g^4$. I understand that $\phi^2$ is like the $\epsilon$ operator of the critical Ising model, and that the temperature or equivalently deviation of $h$ from $J$ corresponds to $m$, however, what do $g$ and $phi^4$ correspond to exactly?