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# categorical direct product

Let $\{C_{i}\}_{{i\in I}}$ be a set of objects in a category $\mathcal{C}$. A *direct product* of the collection $\{C_{i}\}_{{i\in I}}$ is an object $\prod_{{i\in I}}C_{i}$ of $\mathcal{C}$, with morphisms $\pi_{i}\colon\prod_{{j\in I}}C_{j}\to C_{i}$ for each $i\in I$, such that:

For every object $A$ in $\mathcal{C}$, and any collection of morphisms $f_{i}\colon A\to C_{i}$ for every $i\in I$, there exists a unique morphism $f\colon A\to\prod_{{i\in I}}C_{i}$ making the following diagram commute for all $i\in I$.

$\xymatrix{A\ar@{-->}[dr]_{{f}}\ar[rr]^{{f_{i}}}&&C_{i}\\ &\prod_{{j\in I}}C_{j}\ar[ur]_{{\pi_{i}}}}$ |

The morphisms $\pi_{i}\colon\prod_{{j\in I}}C_{j}\to C_{i}$ are called *projection morphisms*.

The direct product of a finite collection of sets $C_{1},C_{2},\ldots,C_{n}$ is often denoted $C_{1}\times C_{2}\times\cdots\times C_{n}$, in analogy with the Cartesian product.

## Mathematics Subject Classification

18A30*no label found*

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