I asked this question on MathStackExchange and was suggested to ask here.

I started studying GIT theory, and I am stuck with the following problem.

Let $\textbf{Sch}$ be the category of schemes over a field (it can be algebraically closed if needed) and $\textbf{Sets}$ be the category of sets. Let $X \in \textbf{Sch}$ and Let $G$ be a algebraic group ($G$ is a group object in $\textbf{Sch}$) acting on $X$.

For each $T \in \textbf{Sch}$, consider the action of $\text{Hom}(T,G)$ on $\text{Hom}(T,X)$: for each morphism $g : T \to G$ and each morphism $x:T \to X$, we have $\text{Hom}(T,G) \times \text{Hom}(T,X) \to \text{Hom}(T,X)$, where $(g,x) \to (g(t), x(t))$. We say that two morphisms $x$, $y : T \to X$ are in the same class if there exist one $g \in \text{Hom}(T,G)$ such that $x(t) = g(t)y(t)$ for all $t$.

Define the functor $\mathcal{F} : \textbf{Sch} \to \textbf{Sets} $ that sends each scheme $T$ to the set of classes of equivalences of $\text{Hom}(T,X)$ defined as before.

I saw in some notes that I am not able to find that is it possible to say that the space of orbits $X/G$ represents the functor $\mathcal{F}$, if $G$ is a reductive group, but I am not able to find such notes, neither prove this fact, any references are welcome.

Thank you.