In many cases, filtered colimits commute with forgetful functors to $\mathbf{Set}$, for example with $\mathbf{CRing} \to \mathbf{Set}$ or $R-\mathbf{Mod} \to \mathbf{Set}$. Is there a slick way of showing this?
I am mainly interested in this because you use this fact for the computation of stalks of a sheaf, and usually this is proven by saying: "check it if you don't believe it"
1) Let $T$ be a monad on a cocomplete category $C$. It is a general and trivial fact that the forgetful functor $\mathsf{Mod}(T) \to C$ preserves all colimits which $T$ presveres. In particular, if $T$ preserves filtered colimits (one then says that $T$ is finitary), then the forgetful functor does so. If $T$ is given by a theory of finite operations, $C$ has products which preserve filtered colimits in each variable, then $T$ is finitary.
2) Stalks of sheaves on $X$ commute with colimits because the stalk functor is left adjoint; in fact it is given by $i^*$ where $i : \{x\} \to X$, with right adjoint $i_*$.