TITLE: A question about co-exponentials
QUESTION [8 upvotes]: An exponential object $B^{A}$ is defined to be the representing object of the functor $$\mathcal{C}\left(- \times A,B\right):  \mathcal{C} \rightarrow Set$$
or equivalently, as the terminal object of $\left(-\times A \downarrow B\right)$. The dual concept is of the co-exponential object which is the initial object of the $\left(B\downarrow -\times A \right)$. 

Is co-exponential object as useful as exponential object? What is the notation for them and what are the interesting examples of co-exponential objects? What is right (or left) adjoint of the functor which send any object to the co-exponential (with a fixed base)?

Thanks

REPLY [2 votes]: I have given an explanation with example and a lot of references about how to think of co-exponentials. 
I have not yet worked out how useful they are for myself under that name.
But a very powerful argument to take them seriously is that reasoning requires the ability to think of how to falsify arguments put forward by someone else. So when someone puts forward an argument in constructive logic to the effect that $\Delta \vdash \alpha$ then it can be questioned by refuting $\alpha$, which in case one accepts the sequent requires one to find one or more statements in $\Gamma$ that one rejects also. This falsificationist reasoning is co-constructive, and that is where co-implications appear too, which is another word for co-exponential. 
It has been forcefully argued that Scientific Reasoning is falsificationist by Karl Popper. In which case it would be very important.