One (Category) is Terminal Object
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Theorem
Let $\mathbf{Cat}$ be the category of categories.
Let $\mathbf 1$ be the category one.
Then $\mathbf 1$ is a terminal object of $\mathbf{Cat}$.
Proof
Let $\mathbf C$ be an object of $\mathbf{Cat}$, i.e. a small category.
From Singleton is Terminal Object of Category of Sets, there exist unique mappings:
- $F_0: \mathbf C_0 \to \mathbf 1_0 = \left\{{*}\right\}$
- $F_1: \mathbf C_1 \to \mathbf 1_1 = \left\{{\operatorname{id}_*}\right\}$
since the latter sets are singletons.
It remains to verify that $F: \mathbf C \to \mathbf 1$ so defined, in fact is a functor.
Trivially, $F$ preserves identity morphisms.
That $F$ has the morphism property follows from:
- $\operatorname{id}_* \circ \operatorname{id}_* = \operatorname{id}_*$
Hence $F: \mathbf C \to \mathbf 1$ constitutes a functor.
The result follows by definition of terminal object.
$\blacksquare$
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.2$: Example $2.11$: $2$