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I've read the statement in the title in a text as a side note without proof.

One way is easy: using Yoneda, terminality of a presheaf $X \in [\mathcal{C}^\text{op}, \textbf{Set}]$ implies that for any other presheaf $F$, so in particular for any representable presheaf $y(C) = \text{Hom}_{\mathcal{C}}(-, C)$, there is exactly one natural transformation $\mu : y(C) \Rightarrow X$, i.e.: $$|X(C)| = |\text{Hom}_{[\mathcal{C}^\text{op}, \textbf{Set}]}(\text{Hom}_{\mathcal{C}}(-, C), X)| = 1. $$ For the converse, Yoneda again quickly guarantees that if all $X(C), \, C \in \mathcal{C}_0$ are singletons, then $X$ is terminal when restricted to representable presheafs. However, I'm not sure how it generalises to arbitrary presheafs.

Concretely, how do I know that if for every $C \in \mathcal{C}_0 = \text{ob}(\mathcal{C})$, there is a unique natural transformation from $y(C)$ to $X$, there is in fact a unique natural transformation from every presheaf $F$ to $X$?

Arctic Char
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Jos van Nieuwman
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  • I hope to avoid concepts like Kan extensions, colimiting cocones, comma/down categories, etc. But perhaps that is too tall an order. – Jos van Nieuwman May 12 '23 at 20:00
  • Just like you use Yoneda for uniqueness, you can use for existence too. Just send every $c \in F(C)$ to the unique $x_C: y(C) \to X$. Uniqueness will give you naturality. – interregno May 12 '23 at 20:20
  • Are you talking about factoring our arbitrary $\mu : F \Rightarrow X$ via $y(C)$ by vertical composition? – Jos van Nieuwman May 12 '23 at 21:42
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    No, I mean as in defining $\mu$ as such: $\mu(c)$ for $c \in F(C)$ as the unique $x \in X(C)$. – interregno May 12 '23 at 21:49

1 Answers1

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A pretty quick way is to observe that every presheaf $F \in [\mathcal{C}^{op}, Set]$ is a colimit of representables in $[\mathcal{C}^{op}, Set]$.

Thus, $[\mathcal{C}^{op}, Set](F, X) \cong [\mathcal{C}^{op}, Set](colim_{i \in I}\text{ }\mathcal{C}(-,U_i), X) \cong lim_{i \in I^{op}}[\mathcal{C}^{op}, Set](\mathcal{C}(-,U_i), X) \cong lim_{i \in I^{op}}XU_i$.

Sincer each $XU_i$ is a singleton (terminal objects in Set), hence the latter limit will also be a terminal object in Set, that is, a singleton.

So $X$ is a terminal object in the presheaf category.

Category_Theorist
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