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Is the following statement correct in the category of sets?

Let $X$ be any set. Then there exists a filtered small category $I$ and a functor $F:I\to \mathrm{Set}$ such that for all $i\in I$ the set $F(i)$ is finite, and such that $$ X \; = \; \mathrm{colim}_{i\in I} F(i) . $$

Are there references on results of this type in the literature?

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The answer is yes: every set is the union of its finite subsets.

So take $I = P_{\text{finite}}(X)$ with as morphisms the inclusion maps, and $F : I \to \text{Set}$ the inclusion.

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One answer already mentions the diagram of finite subsets of $X$. You would have to check that taking the union of this system actually is the colimit (which is an easy exercise).

Since you asked for a reference, Locally Presentable and Accessible Categories by J. Adámek and J. Rosický is a great book on this kind of stuff. In particular example 1.2(1) already mentions the diagram of finite subsets.

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