The representable natural transformation associated to $f$ is the natural transformation
\[ h_{f}\colon h_{A}\Rightarrow h_{B} \]
consisting of the collection
\[ \left\{ h_{f|X} \colon \underbrace{h_{A}(X)}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Hom}}_{\mathcal{C}}(X,A)} \to \underbrace{h_{B}(X)}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Hom}}_{\mathcal{C}}(X,B)} \right\} _{X\in \operatorname {\mathrm{Obj}}(\mathcal{C})} \]
with
\[ h_{f|X} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{*}, \]
where $f_{*}$ is the postcomposition by $f$ morphism of Chapter 11: Categories, Item 2 of Definition 11.1.4.1.1.
