Theorem: Characterization of Existence of Bifunctor Induced By One-Variable Functors/Sufficient Condition

Let $\mathbf C$, $\mathbf D$ and $\mathbf E$ be categories.
For each object $C \in \mathbf C$, let:
:$L_C: \mathbf D \to \mathbf E$ be a covariant functor.
For each object $D \in \mathbf D$, let:
:$M_D: \mathbf C \to \mathbf E$  be a covariant functor.
Let:
:for morphisms $f: C \to C' \in \mathbf C$ and $g: D \to D' \in \mathbf D$:
::$\map {M_{D'} } f \circ \map {L_C} g = \map {L_{C'} } g \circ \map {M_D} f$
Then:
:there exists a bifunctor $F: \mathbf C \times \mathbf D \to \mathbf E$ such that:
::$(1)\quad$ For each object $C \in \mathbf C : F(C, -) = L_C$
::$(2)\quad$ For each object $D \in \mathbf D : F(-, D) = M_D$
:where:
::$F(C, -)$ denotes the functor from $\mathbf D$ to $\mathbf E$ obtained by substitution of first variable with $C$
::$F(-, D)$ denotes the functor from $\mathbf C$ to $\mathbf E$ obtained by substitution of second variable with $D$

Proof:
By hypothesis, for identity morphisms $\operatorname{id}_C : C \to C \in \mathbf C$ and  $\operatorname{id}_D : D \to D \in \mathbf D$:
:$\map {M_D} {\operatorname{id}_C}$ and $\map {L_C} {\operatorname{id}_D}$ are composable with:
::$\map {M_D} {\operatorname{id}_C} \circ \map {L_C} {\operatorname{id}_D} = \map {L_C} {\operatorname{id}_D} \circ \map {M_D} {\operatorname{id}_C}$
For each object $\tuple{C, D} \in \mathbf C \times \mathbf D$, we have:
{{begin-eqn}}
{{eqn | l = \map {M_D} C
      | r = \operatorname{dom} \operatorname{id}_{\map {M_D} C}
      | c = {{Defof|Identity Morphism}}
}}
{{eqn | r = \operatorname{dom} \map {M_D} {\operatorname{id}_C}
      | c = {{Defof|Covariant Functor}}
}}
{{eqn | r = \operatorname{cod} \map {L_C} {\operatorname{id}_D}
      | c = {{Defof|Composite Morphism}}
}}
{{eqn | r = \operatorname{cod} \operatorname{id}_{\map {L_C} D}
      | c = {{Defof|Covariant Functor}}
}}
{{eqn | r = \map {L_C} D
      | c = {{Defof|Identity Morphism}}
}}
{{end-eqn}}
Let $F_0$ be the object functor defined by:
:for each object $\tuple{C, D} \in \mathbf C \times \mathbf D: \map {F_0} {C, D} = \map {M_D} C = \map {L_C} D$
Let $F_1$ be the morphism functor defined by:
:for each morphism $\tuple{f: C \to C', g: D \to D'} \in \mathbf C \times \mathbf D: \map {F_1} {f, g} = \map {M_{D'} } f \circ \map {L_C} g = \map {L_{C'} } g \circ \map {M_D} f$
We have:
{{begin-eqn}}
{{eqn | l = \operatorname{dom} \map {F_1} {f, g}
      | r = \operatorname{dom} \paren {\map {L_{C'} } g \circ \map {M_D} f}
      | c = Definition of $F_1$
}}
{{eqn | r = \operatorname{dom} \map {M_D} f
      | c = {{Defof|Composite Morphism}}
}}
{{eqn | r = \map {M_D} C
      | c = {{Defof|Domain (Category Theory)|Domain of Morphism}}
}}
{{eqn | r = \map {F_0} {C, D}
      | c = Definition of $F_0$
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = \operatorname{cod} \map {F_1} {f, g}
      | r = \operatorname{cod} \paren {\map {L_{C'} } g \circ \map {M_D} f}
      | c = Definition of $F_1$
}}
{{eqn | r = \operatorname{cod} \map {L_{C'} } g
      | c = {{Defof|Composite Morphism}}
}}
{{eqn | r = \map {L_{C'} } {D'}
      | c = {{Defof|Codomain (Category Theory)|Codomain of Morphism}}
}}
{{eqn | r = \map {F_0} {C', D'}
      | c = Definition of $F_0$
}}
{{end-eqn}}
It follows that $F_0$ and $F_1$ are well-defined.
For each $C \in \mathbf C$:
:For each object $D \in \mathbf D : \map {L_C} D = F_0 \tuple{C, D}$
:For each morphism $g \in \mathbf D : \map {L_C} g = F_1 \tuple{\operatorname{id}_C, g}$
For each $D \in \mathbf D$, let:
:For each object $C \in \mathbf C : \map {M_D} C = \map {F_0} {C, D}$
:For each morphism $f \in \mathbf C : \map {M_D} f = \map {F_1} {f, \operatorname{id}_D}$
From Characterization of Bifunctor Induced By One-Variable Functors:
:$F$ is a bifunctor