TITLE: Why are coproduct objects corepresentations of cartesian products, rather than representations of disjoint unions?
QUESTION [5 upvotes]: One way we can define the product of two objects $A$ and $B$ in some category $\mathcal C$ is as a representation of the contravariant functor $(\to A) \times (\to B)$ in $[\mathcal C^{\mathrm{op}}, Set]$.
The analogous definition of coproducts is as a corepresentation of the covariant functor $(A \to) \times (B \to)$ in $[\mathcal C, Set]$.
Notably, AFAICT one can't get products by asking for a corepresentation of $(A \to) + (B \to)$, nor coproducts by asking for a representation of $(\to A) + (\to B)$, where here I'm using $+$ to denote coproducts.
I find this asymmetry curious, and my main question is the title question. That question's a bit broad and vague, though. To be a bit more specific, aspects of this question that I'm particularly curious about include:

What are these other objects? ((co)representations of $(A \to) + (B \to)$) Are they well-studied? Alternatively, are there good reasons to consider them boring and ignore them?
Whence the asymmetry? It's almost as if $Set$ is insisting that $\times$ is "more primary" than $+$. Of course, $Set^{\mathrm{op}}$ will sing the opposite tune, but I still find the situation surprising. Is there a good intuition for why the "correct" way to ask for coproducts in arbitrary categories is equivalent to asking for a corepresentation of Set-theoretic products, rather than (say) a direct representation of Set-theoretic coproducts?
Given a construction in $Set$ (such as a product or a coproduct or an exponential), what determines whether that construction in going to be "representable-ish" versus "corepresentable-ish"? We could of course just try both and see which one works and classify our constructions accordingly, but is there any way to look at products and coproducts in $Set$ and notice ab initio that products are going to be the "representy" ones and coproducts the "corepresenty" ones? (Alternatively, what are the keywords I should be searching to read up on this?)


(lmn if I should expand on my notation more here. Note that I'm following the nlab's convention when distinguishing between representable functors and corepresentable functors, and that by eg $(\to A) \times (\to B)$ I mean $(X \mapsto \mathrm{Hom}_{\mathcal C}(X, A) \times \mathrm{Hom}_{\mathcal C}(X, B))$.)

REPLY [3 votes]: Coproducts don't always look much like disjoint unions; consider the coproduct of groups, namely the free product. More formally, there's a condition called disjointness that coproducts can satisfy, and they don't always satisfy it.
You can think of your definition as an attempt to formalize disjointness; unfortunately it doesn't work. You're asking for an object $D$ such that a morphism $f : C \to D$ is either a morphism $C \to A$ or a morphism $C \to B$. Unfortunately, even disjoint unions don't satisfy this property! You don't expect this kind of thing unless $C$ is connected in some sense (say, it is a connected topological space in $\text{Top}$ or a connected graph in $\text{Graph}$); in general some part of $C$ might map to $A$ and some other part can map to $B$. 
Nevertheless there is something to this idea; it's one way to try formalizing a sum type, namely a type whose terms are either terms of some type $A$ or some other type $B$. From this perspective it's not obvious that sum types have anything to do with coproducts; I wish I understood this better than I do.
A definition that seems related to bridging this gap is the notion of an extensive category.