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 $C \mathop / \mathbf C$.

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

Then there is a morphism $a: \operatorname{id}_C \to f$ iff:


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

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 $\mathbf C$:


 * $\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.