The corepresentable natural transformation associated to $f$ is the natural transformation
\[ h^{f}\colon h^{B}\Rightarrow h^{A} \]
consisting of the collection
\[ \left\{ h^{f}_{X} \colon \underbrace{h^{B}(X)}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Hom}}_{\mathcal{C}}(B,X)} \to \underbrace{h^{A}(X)}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Hom}}_{\mathcal{C}}(A,X)} \right\} _{X\in \operatorname {\mathrm{Obj}}(\mathcal{C})} \]
with
\[ h^{f}_{X} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f^{*}, \]
where $f_{*}$ is the precomposition by $f$ morphism of Chapter 11: Categories, Item 1 of Definition 11.1.4.1.1.
