Product Category is Category
Theorem
Let $\mathbf C$ and $\mathbf D$ be metacategories.
Then the product category $\mathbf C \times \mathbf D$ is a metacategory.
Proof
Let $\left({X,Y}\right), \left({X',Y'}\right) \in \mathbf C \times \mathbf D$.
Let $\left({f,g}\right) : \left({X,Y}\right) \to \left({X',Y'}\right)$ and $\left({h,k}\right) : \left({X',Y'}\right) \to \left({X,Y}\right)$ be morphisms.
Let $\operatorname{id}_X$, $\operatorname{id}_Y$ be the identity morphisms for the objects $X$ and $Y$ respectively.
Then:
| \(\ds \left({f, g}\right) \circ \left({\operatorname{id}_X, \operatorname{id}_Y}\right)\) | \(=\) | \(\ds \left({f \circ \operatorname{id}_X, g \circ \operatorname{id}_Y}\right)\) | By the definition of composition in the product category. | |||||||||||
| \(\ds \) | \(=\) | \(\ds \left({f, g}\right)\) | By the definition of the identity morphisms |
Similarly:
| \(\ds \left({\operatorname{id}_X, \operatorname{id}_Y}\right) \circ \left({h, k}\right)\) | \(=\) | \(\ds \left({\operatorname{id}_X \circ h, \operatorname{id}_Y \circ k}\right)\) | By the definition of composition in the product category. | |||||||||||
| \(\ds \) | \(=\) | \(\ds \left({h, k}\right)\) | By the definition of the identity morphisms |
Therefore, $\left({\operatorname{id}_X, \operatorname{id}_Y}\right)$ satisfies the property of an identity morphism.
Let $\left({f, g}\right)$, $\left({h, k}\right)$ and $\left({\ell, m}\right)$ be composable morphisms of $\mathbf C \times \mathbf D$. We have:
| \(\ds \left({ \left({f, g}\right) \circ \left({h, k}\right) }\right) \circ \left({\ell, m}\right)\) | \(=\) | \(\ds \left({f \circ h, g \circ k}\right) \circ \left({\ell, m}\right)\) | By the definition of composition in the product category. | |||||||||||
| \(\ds \) | \(=\) | \(\ds \left({ \left({f \circ h}\right) \circ \ell, \left({g \circ k}\right) \circ m}\right)\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \left({f \circ \left({h \circ \ell}\right), g \circ \left({k \circ m}\right) }\right)\) | By associativity of morphisms of $\mathbf C$ and $\mathbf D$. | |||||||||||
| \(\ds \) | \(=\) | \(\ds \left({f, g}\right) \circ \left({ \left({h, k}\right) \circ \left({\ell, m}\right) }\right)\) |
Therefore, composition of morphisms in $\mathbf C \times \mathbf D$ is also associative.
$\blacksquare$