In the stacks project website, the proof of lemma 6.27.3, https://stacks.math.columbia.edu/tag/0099 , it claims that the stalk functor $[-]_x:\mathscr{F}\mapsto \mathscr{F}_x$ could be seen as the pullback functor for the inclusion morphism $i_x:\{x\}\to X$ between topological spaces (where $x$ is a point in X), which means
I am confused about this claim. Firstly, the slice category $\mathbf{Top}/X$ consists of morphisms (continuous maps)$Y\to X$, so could each such topological space $Y$ be seen as a sheaf over $X$?
For any continuous morphism of topological spaces $f: Y \to X$, we define the pullback $f^*\mathcal{F}$ of a sheaf $\mathcal{F}$ to be $$f^*\mathcal{F}(U)= \mathrm{colim}_{f(U)\subset V} \mathcal{F}(V)$$ Where $U$ is an open subset of $Y$ and $V$ is an open subset of $X$. If you apply this to the special case $Y=\left \{ x \right \}$ and $f=i_x$ is the inclusion. Then the definition above is reduced to $$i_x^*\mathcal{F}(\left \{ x \right \}) = \mathrm{colim}_{ x \in V} \mathcal{F}(V) = \mathcal{F}_x.$$ The notation $\mathbf{Top}/X$ indicates the cateogory of sheaves on $X$, not the slice category.