Slice Category is Category

Theorem
Let $\mathbf C$ be a metacategory.

Let $C \in \mathbf C_0$ be a object of $\mathbf C$.

Let $\mathbf C / C$ be the slice category of $\mathbf C$ over $C$.

Then $\mathbf C / C$ is a metacategory.

Proof
Let us verify the axioms $(C1)$ up to $(C3)$ for a metacategory.

Suppose that $a: f \to g$ and $b: g \to h$ are morphisms in $\mathbf C / C$.

To show that $b \circ a: f \to h$ is a morphism as well, compute:

Hence $b \circ a: f \to h$ is a morphism.

For $(C2)$, observe that for $a: f \to g$, with objects $f: X \to C$ and $g: Y \to C$, we have:

Hence $(C2)$ is shown to hold.

Since the composition in $\mathbf C / C$ is inherited from $\mathbf C$, satisfaction of the associative property $(C3)$ is also inherited.

Hence $\mathbf C / C$ is a metacategory.