Let be a set of objects in a category . A direct product of the collection is an object of , with morphisms for each , such that:
For every object in , and any collection of morphisms for every , there exists a unique morphism making the following diagram commute for all .
![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ A \ar@{-->}[dr]_{f} \ar[rr]^{f_i} & & C_i \ & \prod_{j \in I} C_j \ar[ur]_{\pi_i} } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/2855/l2h/img14.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ A \ar@{-->}[dr]_{f} \ar[rr]^{f_i} & & C_i \ & \prod_{j \in I} C_j \ar[ur]_{\pi_i} } } \end{xy}$")