Category theory : product and coproduct

[Sumanta]

Hi,

I was trying to understand coproduct and product as defined in category theory from the website

http://en.wikiversity.org/wiki/Introduction_to_Category_Theory/Products_and_Coproducts.

Before I could even think of sth difficult there are some simple things which I don't seem to understand. If anybody could kindly explain it would be helpful.

Not every category has products for all pairs of objects (i.e. 'has all products'). For example in the category with 3 objects and 2 arrows (+identity arrows) shown at right, the product of A and C is the object A together with morphisms.

http://upload.wikimedia.org/wikipedia/commons/f/f8/Simple_category.svg.

I see that t$\pi$$_{1}$g = $\pi$$_{2}$g where the t is the morphism between A and C. But does this somehow fail the uniqueness of g. Not sure how did he get the product as A with the two morphisms identity and t.

Excuse me as not being able to see the Greek letters even after I typed from the latex help

 

[micromass]

The point of that diagram is that A and B does not have a product. AxC does exist and is equal to A.