In Tom Leinster's Basic Category Theory, it says that the coproduct of two representable functors is never representable, but I cannot see why this is true. Suppose we have two Hom-functors $h_a$ and $h_b$ and the coproduct $a+b$ exists. Then, $h_a+h_b$ is to be computed pointwise, i.e. for any object $c$, $(h_a+h_b)(c) = h_a(c)+h_b(c)$, which is just the disjoint union of the Hom-sets from $c$. Now how is this different from $h_{a+b}$? The injections of $a$ and $b$ into the coproduct make $\text{Hom}(c,a)+\text{Hom}(c,b)$ isomorphic to $\text{Hom}(c,a+b)$ as sets, don't they? What am I overlooking here?
Please provide an argument that shows why the coproduct of two representables is never representable.
Assume that for some objects $a,b,c$ of $\mathcal{C}$ we have $h_a+h_b\cong h_c$. Then for any functor $F:\mathcal{C}^{op}\to \mathbf{Set}$, we have by definition of the coproduct $h_a+h_b$ $$\operatorname{Nat}(h_a,F)\times \operatorname{Nat}(h_b,F)\cong \operatorname{Nat}(h_a+h_b,F)\cong \operatorname{Nat}(h_c,F),$$ and then by the Yoneda lemma $$F(a)\times F(b)\cong F(c).$$ But clearly this cannot be true for the functor $F$ that maps every object to the constant set $\{0,1\}$ (or any finite set with more than one element) and every arrow to the identity, so we have a contradiction.