Construction of a function from $\mathbb R^{\mathbb R^2}\to \mathbb R^\Gamma $ where $\Gamma =\mathbb R^2/_\sim$.

Question

Let $\sim$ an equivalence relation, and denote $\Gamma=\mathbb R^2/_\sim$ the quotient space. We denote $\pi:\mathbb R^2\to \Gamma $ the natural projection. And now I want to identify $\mathbb R^\Gamma $ with $\mathbb R^{\mathbb R^2}$ in a natural way.

Is there a natural way map from a set $A$ and $\mathbb R^A$ so that I can define easily the map $\mathbb R^{\Gamma }\to \mathbb R^{\mathbb R^2}$ or the map $\mathbb R^{\mathbb R^2}\to \mathbb R^\Gamma $ ?

Answer 1

$$g\mapsto g\circ\pi$$ serves as prescription of a map: $$\mathbb R^{\Gamma}\to\mathbb R^{\mathbb R^2}$$