The Stacks project

Lemma 19.2.3. Suppose that, in (, $\mathcal{C}$ is the category of sets and $A$ is a finite set, then the map is a bijection.

Proof. Let $f : A \to \mathop{\mathrm{colim}}\nolimits B_\beta $. The range of $f$ is finite, containing say elements $c_1, \ldots , c_ r \in \mathop{\mathrm{colim}}\nolimits B_\beta $. These all come from some elements in $B_\beta $ for $\beta \in \alpha $ large by definition of the colimit. Thus we can define $\widetilde{f} : A \to B_\beta $ lifting $f$ at a finite stage. This proves that ( is surjective. Next, suppose two maps $f : A \to B_\gamma , f' : A \to B_{\gamma '}$ define the same map $A \to \mathop{\mathrm{colim}}\nolimits B_\beta $. Then each of the finitely many elements of $A$ gets sent to the same point in the colimit. By definition of the colimit for sets, there is $\beta \geq \gamma , \gamma '$ such that the finitely many elements of $A$ get sent to the same points in $B_\beta $ under $f$ and $f'$. This proves that ( is injective. $\square$

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