Let $X$ be a topological space (somewhat abusively also let it denote the category of open sets in $X$) and $\mathsf{Set}_X$ the category of $\mathsf{Set}$-valued sheaves over $X$. Let $\mathcal{F},\mathcal{G}$ be sheaves on $X$ and suppose $\varphi:\mathcal{F}\rightarrow\mathcal{G}$ is a map of sheaves. Then there is an induced map $\varphi_p:\mathcal{F}_p\rightarrow\mathcal{G}_p$ on stalks, where $\mathcal{F}_p=\varinjlim\mathcal{F}(U)$, ranging over $U\ni p$, given by the universal property of the colimit (explicitly $\varphi_p(U,s)\mapsto(U,\varphi_U(s))$). In other words, each $p\in X$ yields a functor $\mathsf{Set}_X\rightarrow\mathsf{Set}$ which takes the stalk at $p$.
Stalks seem to be quite important in the study of sheaves and their applications. I'm wondering if there's an equivalent definition for stalks of a sheaf over a locale $X$. I suppose one could start by considering points of a locale, namely maps $*\rightarrow X$ from the two-element locale. Since this is really a map between frames in the opposite direction, we can take the colimit of the $\mathcal{F}(U)$ ranging over the preimage of the top element in the two-element frame.
Is there any way to tell when a functor $\mathsf{Set}_X\rightarrow\mathsf{Set}$ for $X$ a topological space arose from a point $p\in X$ (up to isomorphism)? If the definition for a stalk of a sheaf over a locale I have given is "the correct one", can we similarly characterize the functors which arise from stalks when $X$ is a locale?
I suspect the nlab page on points of topoi is related to what I'm talking about but I really don't know anything about topoi/Grothendieck topoi or geometric morphisms, so hopefully there is a more elementary explanation (especially since I am not asking about topoi in general).