Dual Category is Category

Theorem
Let $\mathbf C$ be a metacategory.

Let $\mathbf C^{\text{op} }$ be its dual category.

Then $\mathbf C^{\text{op} }$ is also a metacategory.

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

Let $f^{\text{op} }: C^{\text{op} } \to D^{\text{op} }$ and $g^{\text{op} }: D^{\text{op} } \to E^{\text{op} }$ be morphisms in $\mathbf C^{\text{op} }$.

Then $f: D \to C$ and $g: E \to D$ are morphisms in $\mathbf C$, and so is $f \circ g: E \to C$.

Therefore, also $g^{\text{op} } \circ f^{\text{op} }: C^{\text{op} } \to E^{\text{op} }$ is a morphism in $\mathbf C^{\text{op} }$, and $(C1)$ is shown to hold.

For $(C2)$, observe that for $f^{\text{op} }: C^{\text{op} } \to D^{\text{op} }$, we have:

Similarly:

Hence $(C2)$ is shown to hold.

To show $(C3)$, reason as follows:

Hence $\mathbf C^{\text{op} }$ is a metacategory.