In the paper “Metric Realization of Fuzzy Simplicial Sets” of David Spivak it takes $I=(0,1)$ as poset and consider it as a category. He gives it a Grothendieck topology induce it from consider $I$ as topological space with topology induced from sets of the form $(0,a)$ with $ain I$.

In the paper, he defines the notation $S((0,a))=S^{geq a}$ for a sheaf $S$. So he also defines:

$S(a)=S^{geq a}-colim_{b>a}S^{geq a}$

So my question is how to understand the definition of $S(a)$. From a cetegorial point of view the colimit is unique up to isomorphism.So diferente sets makes the same colimit, in fact the colimit is determined by its cardanility, as we are working in the category of sets. But this difference depends of the choice of the specific set used to represent the colimit.