We can view a morphism $\Phi: A\times B \to C$, as a family of maps $B\to C$ parametrized by $A$. Recall, that a cross product has naturally defined maps on it (from a space onto the cross product), via its universal property. So in some sense, from a categorical view point, having a product as a domain is non-trivial.
I was wondering what happens with a coproduct: via its universal property, the coproduct is the natural space for functions defined on the coproduct (from the coproduct to another space). What is an interpretation of the non-trivial $\Psi: C \to A \coprod B$?
The only reason it looks like there's a difference here is that you know cases in which $A\times B$ has a universal property both for mapping out and mapping in. In certain categories, such as sets, sheaves, and nice topological spaces, a map $A\times B\to C$ is a map $A\to C^B$. But in general this is not true. $A\times B$ might have a different mapping-out universal property, as in the category of abelian groups. Or, more often, it doesn't have one at all.
Precisely the dual remarks hold for the coproduct. In certain categories such as the opposite of the category of sets, a map $A\to B\sqcup C$ is a map $B^A\to C$, where $B^A$ is an object of homs from $A$ to $B$. They are called cocartesian coclosed, but they essentially only arise as opposites of the more familiar cartesian closed categories, which is why you've never seen such a property before. But formally, the two situations are identical.