Theorem: Characterization of Natural Transformation Between Bifunctors

Let $\mathbf C, \mathbf D$ and $\mathbf E$ be categories.
Let $F, G : \mathbf C \times \mathbf D \to \mathbf E$ be bifunctors from the product category $\mathbf C \times \mathbf D$ to the category $\mathbf E$.
For each object $C \in\mathbf C$ let: 
:$F \tuple{C, -}$ and $G \tuple{C, -}$
denote the functors obtained by substituting the first variable of $F$ and $G$, respectively, with $C$.
For each object $D \in\mathbf D$ let:
:$F \tuple{-, D}$ and $G \tuple{-, D}$
denote the functors obtained by substituting the second variable of $F$ and $G$, respectively, with $D$.
For each object $\tuple{C, D} \in \mathbf C \times \mathbf D$ let:
:$\alpha_{\tuple{C, D}} : F \tuple{C,D} \to G \tuple{C,D}$ 
denote a morphism in $\mathbf E$.
Then:
:$\alpha : F \to G$ is a natural transformation
{{iff}}
:$(1)\quad$ For each object  $C \in \mathbf C : \alpha \tuple{C, -} : F \tuple{C, -} \to G \tuple{C, -}$ is a natural transformation
:$(2)\quad$ For each object  $D \in \mathbf D : \alpha \tuple{-, D} : F \tuple{-, D} \to G \tuple{-, D}$ is a natural transformation
where:
:for objects $C \in \mathbf C$ and $D \in \mathbf D$, $\alpha \tuple{C, -}_D = \alpha_{\tuple{C, D}}$
and:
:for objects $C \in \mathbf C$ and $D \in \mathbf D$, $\alpha \tuple{-, D}_C = \alpha_{\tuple{C, D}}$

Proof:
By definition of natural transformation:
:for all morphisms $f: C \to C' \in \mathbf C$ and $g:D \to D' \in \mathbf D$:
::$\map {G \tuple{-, D} } f \circ \alpha \tuple{-, D}_C = \alpha \tuple{-, D}_{C'} \circ \map {F \tuple{-, D} } f$
::$\map {G \tuple{C, -} } g \circ \alpha \tuple{C, -}_D = \alpha \tuple{C, -}_{D'} \circ \map {F \tuple{C, -} } g$
That is 
:for all morphisms $f: C \to C' \in \mathbf C$ and $g:D \to D' \in \mathbf D$:
::$(3)\quad \map G {f, \operatorname{id}_D} \circ \alpha_{\tuple{C, D} } = \alpha_{\tuple{C', D} } \circ \map G {f, \operatorname{id}_D}$
::$(4)\quad \map G {\operatorname{id}_C, g} \circ \alpha_{\tuple{C, D} } = \alpha_{\tuple{C, D'} } \circ \map F {\operatorname{id}_C, g}$
by definitions of 
:* functors obtained by substituting the first variable and substituting the second variable
:* $\alpha \tuple{C, -}$ and $\alpha \tuple{-, D}$
For all morphisms $\tuple{f: C \to C', g:D \to D'} \in \mathbf C \times \mathbf D$, we have:
{{begin-eqn}}
{{eqn | l = \map G {f, g} \circ \alpha_{\tuple{C, D} }
      | r = \map G {\operatorname{id}_{C'} \circ f, g \circ \operatorname{id}_D} \circ \alpha_{\tuple{C, D} }
      | c = {{Defof|Identity Morphism}}
}}
{{eqn | r = \map G {\tuple{\operatorname{id}_{C'}, g} \circ \tuple{f, \operatorname{id}_D} } \circ \alpha_{\tuple{C, D} }
      | c = {{Defof|Product Category|Composition in Product Category}}
}}
{{eqn | r = \map G {\operatorname{id}_{C'}, g} \circ \map G {f, \operatorname{id}_D} \circ \alpha_{\tuple{C, D} }
      | c = {{Defof|Bifunctor}}
}}
{{eqn | r = \map G {\operatorname{id}_{C'}, g} \circ \alpha_{\tuple{C', D} } \circ \map F {f, \operatorname{id}_D}
      | c = By $(3)$
}}
{{eqn | r = \alpha_{\tuple{C', D'} } \circ \map F {\operatorname{id}_{C'}, g} \circ \map F {f, \operatorname{id}_D}
      | c = By $(4)$
}}
{{eqn | r = \alpha_{\tuple{C', D'} } \circ \map F {\tuple{\operatorname{id}_{C'}, g} \circ \tuple{f, \operatorname{id}_{D} } }
      | c = {{Defof|Bifunctor}}
}}
{{eqn | r = \alpha_{\tuple{C', D'} } \circ \map F {\operatorname{id}_{C'} \circ f, g \circ  \operatorname{id}_{D} }
      | c = {{Defof|Product Category|Composition in Product Category}}
}}
{{eqn | r = \alpha_{\tuple{C', D'} } \circ \map F {f, g}
      | c = {{Defof|Identity Morphism}}
}}
{{end-eqn}}
It follows that $\alpha: F \to G$ is a natural transformation by definition.