Terminology for set-valued functions that are "included" in another function

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I was wondering, if you have $f:A \rightarrow B$ a mapping onto sets, is there appropriate terminology for when, for a subset $C \subseteq A$, you have $g:C \rightarrow D$ that verifies $\forall e \in C$, $g(e) \subseteq f(e)$. If so, is there a proper notation for this?