# Is there a name for this relation on CFGs?

I'm looking for the name (or a name if there isn't one already) of this relation between $G_1=\left<\Sigma_1,\mathcal{N}_1,\mathcal{R}_1,S_1\right>$ and $G_2=\left<\Sigma_2,\mathcal{N}_2,\mathcal{R}_2,S_2\right>$: $$\exists f_\Sigma\in\Sigma_1\times\Sigma_2,f_\mathcal{N}\in\mathcal{N_1}\times\mathcal{N}_2 \text{ surjective functions s.t. }\\ \mathcal{R}_2=\left\{f_\mathcal{N}(N)\rightarrow f(\alpha)\mid N\rightarrow\alpha\in R_1\right\}\text{ and } S_2=f_\mathcal{N}(S_1)$$ where $f$ is the extension of $f_\Sigma$ and $f_\mathcal{N}$ to words ($f\in(\Sigma_1\cup\mathcal{N}_1)^*\times(\Sigma_2\cup\mathcal{N}_2)^*$).

In other words, you can map $G_1$ to $G_2$ (but not necessarily $G_2$ to $G_1$).

If the mapping function were bijective, this would be an isomorphism, but they're not.

What about if only $f_\mathcal{N}$ is surjective, i.e. $f_\Sigma$ is bijective, or even the identity function?

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