\begin{definition}[Definition:Precomposition Mapping]
Let $A, B, C$ be sets.
Let $\operatorname{Hom}(B, C)$ denote the set of all mappings from $B$ to $C$.
Let $f : A \to B$ be a mapping.
The '''precomposition mapping''' $f^* : \operatorname{Hom}(B, C) \to \operatorname{Hom}(A, C)$ is the mapping that sends a mapping $g : B \to C$ to the precomposition $g \circ f$ with $f$.
\end{definition}