Identity Morphism is Initial Object in Coslice Category
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 an initial object in the coslice category $C / \mathbf C$.
Proof
Let $f: C \to D$ be an object of $C / \mathbf C$.
Then there is a morphism $a: \operatorname{id}_C \to f$ if and only if:
- $f = a \circ \operatorname{id}_C = a$
Thus, $f$ itself defines the unique morphism $\operatorname{id}_C \to f$ in $C \mathop / \mathbf C$.
We therefore have the following commutative diagram in $\mathbf C$:
$\quad\quad \begin{xy} <-3em,0em>*+{C} = "X", <3em,0em>*+{D} = "X2", <0em,4em>*+{C} = "C", "X";"X2" **@{--} ?>*@{>} ?*!/^1em/{f}, "C";"X" **@{-} ?>*@{>} ?*!/^.6em/{\operatorname{id}_C}, "C";"X2" **@{-} ?>*@{>} ?<>(.6)*!/_1em/{f'}, \end{xy}$
Hence the result, by definition of initial object.
$\blacksquare$
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.2$: Example $2.11$: $6$