Understanding products and coproducts

I am currently trying to make sense of products and coproducts in different categories, and even in starting with the basic examples (cartesian product and disjoint union in the category of sets) I've found that I don't understand how they meet the definition of a product and coproduct for one of the conditions.

The definition I was given is as follows. Given a functor $F:I\to C$ where $I$ is a discrete category and $C$ is some arbitrary category, a coproduct is an inductive limit of $F$ (a family of maps indexed by $I$ which is an initial object in the category of arrows ($F \to C$), similarly a product is the projective limit in the category ($C \to F$).

For a family of sets $(X_{i})_{i \in I}$, if we define a set $C$ that is their disjoint union, and equip it with a family of embedddings $\iota_{j} :X_{j} \hookrightarrow C$ by $x\mapsto (x,j)$ for each $j\in I$, I can see how it meets the other properties, but I still don't see why this object has to be initial for $(X_{i})_{i \in I}$.

What is it exactly that makes this family of mappings initial?