Source: Categories for the Working Mathematician, second edition by Saunders Mac Lane.
Given a category $C$, a coproduct diagram is a universal arrow from an object $<a, b>$ of $C\times C$ to the diagonal functor $\Delta: C\to C\times C$. It consists of an object $c$ of $C$ and an arrow $<a, b>\to <c, c>$ of $C\times C$; that is, a pair of arrows $i: a\to c$, $j: b\to c$ from $a$ and $b$ to a common codomain $c$. The universal property of this pair reads: for any pair of arrows $f:a\to d$, $g: b\to d$, there is a unique $h: c\to d$ with $f=h\circ i$, $g=h\circ j$. When such a coproduct exists, the object $c$ is necessarily unique (up to isomorphism in $C$) and is written $c=a\coprod b$, called a coproduct object.
The author then claims that the assignment $<f, g>\mapsto h$ is a bijection $C(a, d)\times C(b, d)\cong C(a\coprod b, d)$ natural in $\mathbf{d}$, with inverse $h\mapsto<hi, hj>$.
My Question: I am having a hard time wrapping my head around the boldface part. How does the said bijection define a natural transformation for each $d$ of $C$? In particular, given $d$ of $C$, what are the two functors in the natural transformation so defined? The functors are from which category to which category? And each object of the first category is mapped to which arrow of the second?
There are just way too many missing pieces. Any help would be greatly appreciated.
It is an isomorphism between the two functors $C \to \mathbf{Set}$
$$C(a,-) \times C(b,-) \cong C(a+b,-).$$
For the notation, $C(a,-)$ is the (covariant) hom-functor, and if $F,G$ are two $\mathbf{Set}$-valued functors, $F \times G$ is the functor defined by $(F \times G)(x)=F(x) \times G(x)$ for objects and a similar formula for morphisms.
A coproduct is really just a representation of the functor $C(a,-) \times C(b,-)$. And more generally, a colimit $\mathrm{colim}_i a_i$ is really just a representation of the functor $\lim_i C(a_i,-)$.