I'm wondering about the deduction of products and coproducts in $\bf Set$. They are both fairly simple objects, yet I cant find any constructive way to deduce them.
Please help
Thanks in advance,
Tobias
Edit 1: Sorry for the vague question. I have a rather specific notion in mind. We can formulate the product of $J$ sets $A_j$ as the set of all maps $f:J\rightarrow \bigcup_{j\in J}A_j$ such that $f(j)\in A_j$ for all $j\in J$ with projections $p_i(f) = f(i)$
Dually, the coproduct can be formulated the union $\bigcup_{j\in J}(g:A_j\rightarrow J)$ such that $g(a)=j, \forall a\in A_j$ with injections $i_j(a) = (a,j)\in g $ for some $g$ in the coproduct
I like these formulations as they display the duality, are based on maps and seem constructive.
What I'd like to know is how to motivate their existence in the comma categories $(\bf{C}\overset{\triangle}{\rightarrow}\bf {C^J}\leftarrow \bf1)$ and its dual.
It feels like it should be simple, yet i've been stuck here for quite some time.
Edit 2: Motivation by adjunction would also be acceptable
Edit 3: So to clarify, I would like to motivate formulating the products and coproducts in $\bf Set$ as above by arguments based on objects in the comma category.
Again, apologies for not being clearer
Edit 4: Fixed definitions so others wondering the same might read it