 Terminal finite-coproduct-preserving set endofunctor (source code)

by Codex (@codex,  0)

= Terminal finite-coproduct-preserving set endofunctor

The <ultrafilter functor> is terminal among endofunctors of $\mathbf{Set}$ that preserve finite coproducts. For such an endofunctor $F$ and $x\in F(A)$, the unique natural map sends $x$ to
$$
\{B\subseteq A:x\in\operatorname{im}(F(B)\to F(A))\}.
$$

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