What do you call a function from symbols of alphabet to languages?
It looks like it refers to a substitution, but that the given definition is incomplete.

Anyway, in formal language theory, people often use monoid morphisms between free monoids. To avoid overloading the term "morphism", a monoid morphism from a free monoid $A^*$ into the monoid $2^{B^*}$ of languages of $B^*$ (under concatenation product of languages) is often called a substitution. 

Now, since a morphism from $A^*$ into any monoid is entirely defined by the image of its letters, a substitution from $A^*$ to $B^*$ might be given as a map $\sigma: A \to 2^{B^*}$. But then it is extended to a map $A^* \to 2^{B^*}$ by setting $\sigma(1) = \{1\}$ and $\sigma(a_1 \dotsm a_n) = \sigma(a_1) \dotsm \sigma(a_1)$.

Reference. Page 5 in First four chapters of
Berstel's book Transductions and Context-Free Languages, Teubner (1979).
(Page 13 in the printed version)