Coproducts in a poset considered as a category. This is a homework question.
We have defined products in a category over an index set $I$ as follows: "A (possibly infinite) product in a category $\mathbb C$ is a limit
of a diagram whose shape is a discrete category."
I could prove that prodcuts in a poset correspond to greatest lower bounds if they exist. Now the following question is:
"Coproducts are defined in the dual way. Show that if a poset 
has products it has coproducts. 
".
But for me that says: If a poset has always greatest lower bounds then also smallest upper bounds.
But that seems for me not to be true since $(\mathbb N,+)$ is a poset and every subset of $\mathbb N$ has a greatest lower bound, but no smallest uppber bound, i.e. $\mathbb N$ itself. 
Can the stated question be true, or do I misunderstand the question ?
 A: The trick here is that you want to construct the least upper bound (LUB) of a set $X$ by invoking the fact your poset has greatest lower bounds (GLB). The easiest way to do that is to find a set $Y$ whose GLB is actually the LUB of $X$.
There is a natural candidate for such a $Y$: the set of all things that "ought" to be greater than or equal to the LUB of $X$. More precisely, let $Y$ be the set of all upper bounds of $X$. It turns out the GLB of this set is also an upper bound of $X$, and thus it is the least upper bound.
This trick is actually useful in greater generality, in relation to "universal arrows" and related topics like limits, colimits and adjoints.
For example, how might we define the coproduct of $X$ and $Y$ in an arbitrary category? We could define the category of all diagrams $X \to Z \leftarrow Y$ (a morphism of diagrams is an arrow $Z \to Z'$ that makes the two resulting triangles commute). A coproduct diagram is then defined to be an initial object in this category.
However, it is an interesting fact an initial object of $\mathscr{C}$, which is the colimit of the empty diagram, also turns out to be the limit of the diagram consisting of the entirety of $\mathscr{C}$. So we can actually define the coproduct by taking an appropriate limit in an appropriate category! (note that if $\mathscr{C}$ is not a small category, then this is not the limit of a small diagram!)
This ability to use limits to compute colimits and vice versa can be useful on occasion (e.g. when adjoints are involved). Unfortunately, I cannot recall any concrete examples.
