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Limits
Products are one example of what is known in category theory as limits. A limit is an object that summarizes a structure (also called a diagram) consisting of other objects and morphisms in a way that allows us to later retrieve some of it.
A limit also has to be unique in the sense that you cannot have two limit objects for the same structure.
The notion of a limit is strongly related with the notion of a commuting diagrams. This is not obvious when we are examining products because the diagram of products does not have several routes reaching to the same point.
Limits can be defined formally, just like everything else that we examine, but we won't bother to do that here.
Products are Limits
OK, we said that limits summarize a structure. What is the structure that a product is summarizing? It is a structure that consists of two objects (sets) that are have no connections between them.
Why is the product unique when it comes to representing the two objects? Because any other object that also represents them is connected to the product through a morphism (this is known as the universal property of limits).