The context of this question is given below but I don't think it is of essence here, so I will try to formulate the question for maps between sets.


Given two sets $A$ and $B$, denote the set of all maps $f\colon A\to B$ by $B^A$. Then, given a set $C$, a map $h\colon B\to C$ "lifts" to a map $h^A\colon B^A\to C^A$ defined by $h^A(f)(a)=h(f(a))$ for all $f\in B^A$ and $a\in A$. 


This construct allows to "change the final codomain" in a "nested" map $F\colon X\to Z^Y$ with a simple composition operation: given a map $H\colon Z\to W$, the composition $H^Y\circ F$ is well defined and creates a map $X\to W^Y$ where the "final codomain" $Z$ has been changed to $W$. The point here is that $H^Y\circ F$ still only takes a single argument from $X$ and after evaluation then a single argument from $Y$. 


In case you are curious, I want to use something like this to calculate with Hessian operators for vector-valued maps on Riemannian manifolds and their duals without using explicit basis notation. The above construct allows to do that quite conveniently, including certain types of "chain rules". 


The questions are: does this construct have a name? Maybe something from category theory? Is this idea of "lifting" $h$ to $h^A$ standard? Does it have a name?



Terminology: changing the codomain in nested maps (e.g. in multilinear algebra)
I think you want to say something like the following:


Let $J,K:{\cal Sets}\times{\cal Sets}\times{\cal Sets}\rightarrow{\cal Sets}$
be defined by 
$$J(X,Y,Z)=Hom(X,Hom(Y,Z))$$
$$K(X,Y,Z)=Nat(Hom(Z,-),Hom(X,Hom(Y,-))$$
where "Nat" means the set of natural transformations between two functors. 


(These will be contravariant in the first two variables and covariant in the third.)


Then the map $F\mapsto H^Y\circ F$ defines a natural transformation from $J$ to $K$.


This contains a bit more content because it says that your construction commutes with compositions with maps into and out of $X$, $Y$ and $Z$. 


