Theorem: Characterization of Hom Bifunctor with Right Functor as Composition of Functors

Let $\mathbf {Set}$ be the category of sets.
Let $\mathbf C$ be a locally small category.
Let $\mathbf D$ be a locally small category.
Let $R : \mathbf D \to \mathbf C$ be a covariant functor.
Let $\map {\operatorname{Hom}_{\mathbf D} } {-, R-} : \mathbf D^{\text{op} } \times \mathbf C \to \mathbf {Set}$ denote the hom bifunctor with right functor.
Then:
:$\map {\operatorname{Hom}_{\mathbf D} } {-, R-} = \map {\operatorname{Hom}_{\mathbf D} } {-, -} \circ \paren{\operatorname{id}_{\mathbf D^\text{op}} \times R}$
where:
:* $\operatorname{id}_{\mathbf D^\text{op}}$ is the identity functor on $\mathbf C^\text{op}$
:* $\operatorname{id}_{\mathbf D^\text{op}} \times R$ denotes the product of functors $\operatorname{id}_{\mathbf D^\text{op}}$ and $R$
:* $\map {\operatorname{Hom}_{\mathbf D} } {-, -}$ denotes the hom bifunctor
:* $\map {\operatorname{Hom}_{\mathbf D} } {-, -} \circ \paren{\operatorname{id}_{\mathbf D^\text{op}} \times R}$ is the composition of $\paren{\operatorname{id}_{\mathbf D^\text{op}} \times R}$ with $\map {\operatorname{Hom}_{\mathbf D} } {-, -}$

Proof:
Let $F$ denote the hom bifunctor with right functor $\map {\operatorname{Hom}_{\mathbf D} } {-, R-} : \mathbf D^\text{op} \times \mathbf C \to \mathbf {Set}$.
Let $G$ denote the hom bifunctor $\map {\operatorname{Hom}_{\mathbf D} } {-, -} : \mathbf D^\text{op} \times \mathbf C \to \mathbf {Set}$.
For each object $\tuple{D^\text{op}, C} \in \mathbf D^{\text{op} } \times \mathbf C$ we have:
{{begin-eqn}}
{{eqn | l = \map {\paren{G \circ \paren{\operatorname{id}_{\mathbf D^\text{op} } \times R} } } {D^\text{op}, C}
      | r = \map G {\map {\paren{\operatorname{id}_{\mathbf D^\text{op} } \times R} } {D^\text{op}, C} }
      | c = {{Defof|Composite Functor|Composition of Functors}}
}}
{{eqn | r = \map G {\map {\operatorname{id}_{\mathbf D^\text{op} } } {D^\text{op} }, R C }
      | c = {{Defof|Product of Functors}}
}}
{{eqn | r = \map G {D^\text{op}, R C}
      | c = {{Defof|Identity Functor}}
}}
{{eqn | r = \map {\operatorname{Hom}_{\mathbf D} } {D, R C}
      | c = {{Defof|Hom Bifunctor}}
}}
{{eqn | r = \map F {D^\text{op}, C}
      | c = {{Defof|Hom Bifunctor With Right Functor}}
}}
{{end-eqn}}
For each morphism $\tuple{f^\text{op} : D_1 \to D_2, g: C_1 \to C_2} \in \mathbf D^{\text{op} } \times \mathbf C$ we have:
{{begin-eqn}}
{{eqn | l = \map {\paren{G \circ \paren{\operatorname{id}_{\mathbf D^\text{op} } \times R} } } {f^\text{op}, g}
      | r = \map G {\map {\paren{\operatorname{id}_{\mathbf D^\text{op} } \times R} } { f^\text{op}, g} }
      | c = {{Defof|Composite Functor|Composition of Functors}}
}}
{{eqn | r = \map G {\map {\operatorname{id}_{\mathbf D^\text{op} } } {f^\text{op} }, R g }
      | c = {{Defof|Product of Functors}}
}}
{{eqn | r = \map G {f^\text{op}, R g}
      | c = {{Defof|Identity Functor}}
}}
{{eqn | r = \map {\operatorname{Hom}_{\mathbf D} } {f, R g} : \map {\operatorname{Hom}_{\mathbf D} } {D_1, R C_1} \to \map {\operatorname{Hom}_{\mathbf D} } {D_2, R C_2} : h \mapsto R g \circ h \circ L f
      | c = {{Defof|Hom Bifunctor}}
}}
{{eqn | r = \map F {f^\text{op}, g}
      | c = {{Defof|Hom Bifunctor With Right Functor}}
}}
{{end-eqn}}
The result follows.
{{qed}}