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Assume that we have a pair of functors $Y:A \to B$ and $F:A \to C$ where $A$ is an essentially small category, $B,C$ are cocomplete categories and $Y,F$ preserve colimits. Assume also that for some ...

In Categories for Working Mathematician, Mac Lane describe a cartesian product as a limit for a functor $F$ from a discrete category $|J|$ : Any cone from an object $Z$ to $F$, is a collection of ...