Is there a natural reason for defining the compactopen topology on the set of continuous functions between two locally compact spaces. For example "to make ... functions continuous". Or in another way of asking this, is there an adjoint functor of the functor, say F, which assigns the topological space $F(X,Y):=Hom_C(X,Y)$ (with the compactopen topology on it) to the couple X,Y.

In regard to your question I recommend Topologies on spaces of continuous functions, Topology Proceedings, volume 26, number 2, pp. 545564, 20012002 by Martin Escardo and Reinhold Heckmann. 


I always liked the following reason: Let's call a topology on a space "admissible" if the evaluation function $e: Hom(X,Y) \times X \rightarrow Y$ is continuous. Then the compactopen topology is coarser than any other admissible topology. In particular, in any case where the compactopen topology is admissible, it is the smallest possible topology that does this. EDIT: See comments for some references. I don't claim any originality here :) 


Exactly as you say, adjoint functors are the answer! (Or at least, they're one possible answer.) In particular, for reasonable spaces $X,Y,Z$, there is a natural isomorphism
where $[Y,Z]$ denotes $\mathrm{Hom}(Y,Z)$ with the compactopen topology. This is exactly the categorical characterisation of an exponential object. This certainly holds when $X,Y,Z$ are compactlygenerated Hausdorff spaces, so the category of such spaces is Cartesian closed. Re Mariano's comment: yes, in some sense this is just fancy language for “things we want to converge, converge; things we don't, don't”. But I think this helps explain why we want the things we want to converge, to converge. ☺ 

