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.