# Definition:Precomposition Mapping

## Definition

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 $f$.