Right now I am reading "Topoi: The Categorial Analysis of Logic", by Goldblatt. I am at the section where he explains the concept of products, by first looking at an example from Set (not using specific sets, just an abstract A and B) and then defining what a product in a category is.
So in getting the concept down I figured I would do actual examples and even non-examples myself. His second example is that in Grp, the category of groups, a product is the direct group product of groups, the binary operation done component-wise. He doesn't use specific groups, just says that the direct group product is a categorial product.
Well I figured that $Z_8$, not being isomorphic to $Z_2$ X $Z_4$ in Grp (though as merely sets are bijective), would be a great non-example of a product (since products are all merely isomorphic anyway they form a class of objects, right? so there might be more than one of them?) I had trouble doing commutative diagrams in latex but this was my set up:
$pr_2\colon Z_2$ X $Z_4 \to Z_2$
$pr_4\colon Z_2$ X $Z_4 \to Z_4$
And here, my "auxilary" object and its homomorphisms:
$f\colon Z_8 \to Z_2$
$g\colon Z_8 \to Z_4$
$(f,g)\colon Z_8 \to Z_2$ X $Z_4$
Where f could either be the trivial homomorphism or x mod 2, and similarly for g.
But no matter what I fill in for f and g, (f,g) seems to not only be defined and commute the diagram, but unique. However, given the group homomorphisms from $[Z_8, Z_2]$ and $[Z_8, Z_4]$, (f,g) is clearly not an isomorphism even if we were to forget the group structure, but not iso even as a homomorphism, yet the product arrow is supposed to be iso.
When first coming to the concept from the book days ago, I was confused if whether or not the auxilary object too was a product but I was confident that I understood that it too is a product, and confident that I was following the logic of the author's text in the language of category theory just by following the "mechanical manipulation of symbols", for example on the next page he proves that all product objects are iso to each other and I felt I was able to follow his argument.
I guess my question is, what am I not understanding in the definition of a product in category theory? How does it entail the product from specific categories such as Grp and Top, and further properties like the product arrows being iso's in the specific categories? And how would that explanation/understanding correct whatever mistake I made with the above example?
Thanks in advance.