# how to understand that products and coproducts are dual

I am reading some basic category notes, how can one relate the products to coproducts? If given a product, can one build its dual product? for example, the coordinate product $(x,y)$ : $R \times R$, what coproduct does it correspond to?

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Note that if $X$ is a product of $A$ and $B$, there may exist a $C$ and $D$ with $C\times D\cong X$ such that the coproduct of $A$ and $B$ is not the same as the coproduct of $C$ and $D$. – Dan Rust Dec 19 '13 at 19:04

Coproduct is the dual of product in the categorical sense, i.e. we have to reverse all occuring arrow in the definition.

While product in most concrete categories of structures is indeed realized on the Cartesian product of the underlying sets (because the forgetful functor usually admits a left adjoint), the coproduct may vary from category to category.

In the simplest cases (e.g. in $\Bbb{Set}$) the coproduct is just the disjoint union. But e.g. in the category of groups, the coproduct is the free product...

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Products and coproducts are dual in the following sense:

Let $\mathcal{C}$ be a category and $\{X_i\}$ be a family of objects in $\mathcal{C}$. Let $\{X'_i\}$ be the same family but considered as a family of objects in $\mathcal{C}^{op}$. Then:

The product $\prod_i X_i$ in $\mathcal{C}$ is the same as the coproduct $\coprod_i X'_i$ in $\mathcal{C}^{op}$.

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