The category $\mathsf{Stone}$ has cofiltered limits Is there an easy way to prove that the category $\mathsf{Stone}$ of Stone spaces and continuous functions has cofiltered limits? Maybe using the fact that the forgetful functor $\mathsf{Stone} \to \mathsf{Set}$ has an adjoint?

The category $\mathsf{Stone}$ is equivalent to the opposite category $\mathsf{Bool}^{op}$ of the category of Boolean algebras. Since $\mathsf{Bool}$ is the category of models of an algebraic theory, it has all limits and colimits, so its opposite category also has all limits and colimits.

Alternatively, $\mathsf{Stone}$ is a reflective subcategory of $\mathsf{Top}$, the reflector taking a space $X$ to the closure of the image of the natural map $X\to \\{0,1\\}^S$ where $S$ is the set of all continuous maps $X\to\\{0,1\\}$. So limits in $\mathsf{Stone}$ are the same as limits in $\mathsf{Top}$ and colimits in $\mathsf{Stone}$ can be computed by taking the colimit in $\mathsf{Top}$ and then applying the reflector.