I'm trying to understand whether there's a fully faithful functor $LRS \supset FormalSch \to IndSch$ and in what sense. Here's my progress so far:

Let $\mathsf{A}$ be the category of adic rings. The objects are topological rings whose topology is generated by a descending filtration of ideals whose intersection is $\{0\}$. Morphisms are continuous homomorphism of rings.

There's a functor $Spf: \mathsf{A} \to \mathsf{IndSch}$ which takes an adic ring to the formal spectrum which is naturally a filtered colimit of (affine) schemes). The target of the functor could be that of *adic* locally ringed spaces (topological spaces with sheaves of adic rings and morphisms between for which the comorphism of sheaves is continuous). Denote this category $\mathsf{ALRS}$.

In $\mathsf{ALRS}$ We have an adjunction with the "continuous" global section functor $\Gamma_{cont} \dashv Spf $. Continuous here just means it remembers the topology (i.e. the filtration).

Now the definition of formal schemes feels inevitable:

Definition:Aformal schemeis an adic locally ringed space locally isomorphic to a formal spectrum of an adic ring. Denote the subcategory of formal schemes by $\mathsf{FSch}\subset \mathsf{ALRS}$.

This raises a problem though. There's no obvious way to turn a "formal scheme" in this sense into an ind-schemes (which are much more convenient for certain purposes). We could try to define the ind-scheme as the formal colimit over the cech nerve of a chosen covering by formal spectra (which are themselves filtered colimits of affine schemes). However, this is probably a very bad idea since it will most likely depend on the choice of covering.

Question:Can we construct a functor $\mathsf{FSch} \to IndSch$ with some good properties? (hopefully fully faithfull but if not maybe at least full?). If not is there a better definition of a formal scheme which enables you to play in both worlds? (ind schemes and locally ringed spaces).