Let $Y$ be a scheme and $y$ be a point of $Y$ with residue field $k(y) = \mathcal{O}_{Y, y}/\mathfrak{m}_{Y, y}$. I came across a claim that $\text{Spec } k(y)$ is the projective limit of the system of open affine neighborhoods of $y$. I understand that $\mathcal{O}_{Y, y}$ can be taken as a direct limit of coordinate rings of affine open neighborhoods of $y$, but I don't understand why this should be true for $k(y)$. I'm not sure if this is relevant, but $Y$ comes with a morphism of finite presentation $f : X \longrightarrow Y$ over some scheme $S$. (https://stacks.math.columbia.edu/tag/01TO)
My first attempt at showing that $k(y)$ is a direct limit was something like the suggestion given in the first paragraph of this question: Structure sheaf of a fiber
However, the answer indicates that this suffers from the same problem I worried about earlier. Is there a way to fix this and write $k(y)$ as a direct limit?