Is I consider an ind scheme such as $G(k((t)))$ for a reductive connected group over $k=\bar{k}$

I have the conjugacy action of $G(k[[t]])$.

In what category can I make the quotient $[G(k((t))/ad(G(k[[t]])]$?

In the category of presheaves? The category of fppf sheaves?

And if I want to make the fiber product of two such object over $k$, same question.

Moreover, if $f:X\rightarrow Y$ is a morphism of presheaves, i.e. contravariant functors from the category of schemes to sets, is the notion of formall smoothness (by using the infinitesimal lifting property as the definition of formal smoothness) stable under base change?