nLab
function set

Given sets $A$ and $B$, the function set $A^B$ is the set of all functions from $B$ to $A$. In the foundations of mathematics, the existence of such a set may be taken to follow from the existence of power sets, from the axiom of subset collection, or as an axiom in its own right.

Thinking of Set as a locally small category, this is a special case of a hom-set. Thinking of $Set$ as a cartesian closed category, this is a special case of an exponential object, which is a special case of an internal hom.

category: foundational axiom