# Definition:Product (Category Theory)/General Definition

## Definition

Let $\mathbf C$ be a metacategory.

Let $\CC$ be any collection of objects of $\mathbf C$.

Let $\map {\mathbf {Dis} } \CC$ be the discrete category on $\CC$, considered as a subcategory of $\mathbf C$.

A product for $\CC$, denoted $\ds \prod \CC$, is a limit for the inclusion functor $D: \map {\mathbf {Dis} } \CC \to \mathbf C$, considered as a diagram.

For an object $C$ in $\CC$, the associated morphism $\ds \prod \CC \to C$ is denoted $\pr_C$ and called the projection on $C$.

The whole construction is pictured in the following commutative diagram:

$\begin{xy}\[email protected]@L+3px{ & & A \ar@{-->}[dd] \ar[dddl]_*+{a_C} \ar[dddr]^*+{a_C'} \\ \\ & & \ds \prod \CC \ar[dl]^*{\pr_C} \ar[dr]_*{\pr_{C'}} \\ \map {\mathbf {Dis} } \CC & C & \dots \quad \dots & C' }\end{xy}$

### Finite Product

Let $\ds \prod \CC$ be a product for a finite set $\CC$ of objects of $\mathbf C$.

Then $\ds \prod \CC$ is called a finite product.

## Also known as

If $\CC = \family {C_i}_{i \mathop \in I}$ is a set, indexed by some indexing set $I$, the notations $\ds \prod_{i \mathop \in I} C_i$ and $\ds \prod_i C_i$ are often seen.

In this situation, one writes $\pr_i$ for $\pr_{C_i}$ and calls it the $i$th projection.