Let $f\colon A\to B$ and $g\colon B\to C$ be morphisms of $\mathcal{C}$.
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The precomposition function associated to $f$ is the function
\[ f^{*} \colon \textup{Hom}_{\mathcal{C}}\webleft (B,C\webright ) \to \textup{Hom}_{\mathcal{C}}\webleft (A,C\webright ) \]
defined by
\[ f^{*}\webleft (\phi \webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\phi \circ f \]for each $\phi \in \textup{Hom}_{\mathcal{C}}\webleft (B,C\webright )$.
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The postcomposition function associated to $g$ is the function
\[ g_{*} \colon \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Hom}_{\mathcal{C}}\webleft (A,C\webright ) \]
defined by
\[ g_{*}\webleft (\phi \webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\circ \phi \]for each $\phi \in \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )$.
