Every functor $F\colon \mathcal{C}\to \mathcal{D}$ defines a natural transformation1
\[ \webleft\{ F^{\dagger }_{A,B} \colon \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Hom}_{\mathcal{D}}\webleft (F_{A},F_{B}\webright ) \webright\} _{\webleft (A,B\webright )\in \text{Obj}\webleft (\mathcal{C}^{\mathsf{op}}\times \mathcal{C}\webright )} \]
with
\[ F^{\dagger }_{A,B} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}F_{A,B}. \]
