Why do we define product of morphisms in this way

Question

I'm always forget the definition of product of morphisms in a category, maybe the main reason is because I don't know the motivation beyond the definition:

I need help to see this abstract definition in a more intuitive way.

Thanks in advance

Answer 1

One good way to think about this is in terms of representable functors: try showing that this definition is equivalent to one reading

"An object $A$ is a product $\prod_{i\in I} A_i$ iff the functor $\hom(\bullet,A)$ is naturally isomorphic to $\prod_{i\in I}\hom(\bullet A_i)$.

The naturality of the isomorphism isn't initially the crux of the matter-although it is important, it follows from uniqueness: the point is really just that maps into $\prod A_i$ are the product of maps into the $A_i$.

This is the sense in which a categorical product generalizes a set-theoretic product, since to specify a map into a product $\prod A_i$ of sets is equivalent to specifying a map into $A_i$ for each $i\in I$. Observe that it really is a generalization since among the maps into a set $S$ are maps $\{*\}\to S$ specifying each point of $S$!

A last thing is that if that once you understand the two cases of the binary product $A\times A'$ and the nullary product $1$ there's no more content to the infinite case, so feel free to get rid of all the $I$s.