I know that it may not always be possible to construct a "product object" for any two objects in a given category. That is not my question. My question is specifically about categories being "multiplied" to give a new category.
2026-05-06 05:17:20.1778044640
Is it always possible to construct a "product category" given any two arbitrary categories C and D?
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The product of two categories $\mathcal C$ and $\mathcal D$ has as its objects the pairs $(C,D)$ such that $C$ is an object of $\mathcal C$ and $D$ is an object of $\mathcal D$. The morphisms from $(C,D)$ to $(C',D')$ in the product category are pairs $(f,g)$ such that $f:C\to C'$ in $\mathcal C$ and $g:D\to D'$ in $\mathcal D$. Morphisms are composed componentwise.