Extending function to a set of object in its domain

Question

Let $f$ be a function from $\mathcal{A}$ to $\mathcal{B}$. I need to extend the function $f$ (to a new function $\widetilde{f}$) from a single object $a \in \mathcal{A}$ to a set of object $A = \{\, a_1, \ldots a_n \,\} \in \mathcal{P}(\mathcal{A})$ such that $\widetilde{f}(A) = \{\, f(a_1), \ldots, f(a_n) \,\}$.

Is there a precise term used in literature to express this concept?

Answer 1

Some candidates are the "image map", "direct image map", or "forward image map" under $f$, or the "power set functor" or "covariant power set functor" applied to $f$. You can find attestations for all of these; I won't pretend to know which is the most popular.