Identity Morphism is Terminal Object in Slice Category

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Theorem

Let $\mathbf C$ be a metacategory, and let $C \in \mathbf C_0$ be an object of $\mathbf C$.

Let $\operatorname{id}_C: C \to C$ be the identity morphism for $C$.


Then $\operatorname{id}_C$ is a terminal object in the slice category $\mathbf C \mathop / C$.


Proof

Let $f: B \to C$ be an object of $\mathbf C \mathop / C$.

Then there is a morphism $a: f \to \operatorname{id}_C$ if and only if:

$f = \operatorname{id}_C \circ a = a$

Thus, $f$ itself defines the unique morphism $f \to \operatorname{id}_C$ in $\mathbf C \mathop / C$.

We therefore have the following commutative diagram in $\mathbf C$:

$\quad\quad \begin{xy} <-3em,0em>*+{B} = "X", <3em,0em>*+{C} = "X2", <0em,-4em>*+{C} = "C", "X";"X2" **@{--} ?>*@{>} ?*!/_1em/{f}, "X";"C" **@{-} ?>*@{>} ?<>(.3)*!/^1em/{f}, "X2";"C" **@{-} ?>*@{>} ?<>(.3)*!/_1em/{\operatorname{id}_C}, \end{xy}$

Hence the result, by definition of terminal object.

$\blacksquare$


Sources

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