Product Category is a Category

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Theorem

Let $\mathcal C$ and $\mathcal D$ be categories.

Then the product category $\mathcal C \times \mathcal D$ is a category.

Let $(X,Y),(X',Y') \in \mathcal C \times \mathcal D$.

Let $(f,g) : (X,Y) \to (X',Y')$ and $(h,k) : (X',Y') \to (X,Y)$ be morphisms.

Let $\operatorname{id}_X$, $\operatorname{id}_Y$ be the identity morphisms for the objects $X$ and $Y$ respectively.

Then:

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle (f,g) \circ (\operatorname{id}_X,\operatorname{id}_Y)$	$=$	$\displaystyle (f\circ \operatorname{id}_X, g\circ \operatorname{id}_Y)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	By the definition of composition in the product category.
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle (f, g)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	By the definition of the identity morphisms

Similarly,

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle (\operatorname{id}_X,\operatorname{id}_Y) \circ (h,k)$	$=$	$\displaystyle (\operatorname{id}_X\circ h, \operatorname{id}_Y \circ k)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	By the definition of composition in the product category.
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle (h, k)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	By the definition of the identity morphisms

Therefore $(\operatorname{id}_X,\operatorname{id}_Y)$ satisfies the property of an identity morphism.

Now let $(f,g)$, $(h,k)$ and $(\ell,m)$ be morphisms of $\mathcal C \times \mathcal D$. We have:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \left[ (f,g) \circ (h,k) \right] \circ (\ell,m)$	$=$	$\displaystyle (f\circ h, g\circ k) \circ (\ell,m)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	By the definition of composition in the product category.
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle (\left[f\circ h\right] \circ \ell, \left[ g\circ k \right] \circ m)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle (f\circ\left[ h \circ \ell\right], g\circ \left[k \circ m\right])$	$\displaystyle $	$\displaystyle $	$\displaystyle $	By associativity of morphisms of $\mathcal C$ and $\mathcal D$.
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle (f,g) \circ \left[ (h,k) \circ (\ell,m)\right]$	$\displaystyle $	$\displaystyle $	$\displaystyle $

Therefore composition of morphisms in $\mathcal C \times \mathcal D$ is also associative.

$\blacksquare$

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle (f,g) \circ (\operatorname{id}_X,\operatorname{id}_Y)\)	\(=\)	\(\displaystyle (f\circ \operatorname{id}_X, g\circ \operatorname{id}_Y)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	By the definition of composition in the product category.
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle (f, g)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	By the definition of the identity morphisms

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle (\operatorname{id}_X,\operatorname{id}_Y) \circ (h,k)\)	\(=\)	\(\displaystyle (\operatorname{id}_X\circ h, \operatorname{id}_Y \circ k)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	By the definition of composition in the product category.
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle (h, k)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	By the definition of the identity morphisms

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left[ (f,g) \circ (h,k) \right] \circ (\ell,m)\)	\(=\)	\(\displaystyle (f\circ h, g\circ k) \circ (\ell,m)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	By the definition of composition in the product category.
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle (\left[f\circ h\right] \circ \ell, \left[ g\circ k \right] \circ m)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle (f\circ\left[ h \circ \ell\right], g\circ \left[k \circ m\right])\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	By associativity of morphisms of $\mathcal C$ and $\mathcal D$.
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle (f,g) \circ \left[ (h,k) \circ (\ell,m)\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)