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?