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\homework{Security}

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\begin{exercise}
Prove that the 2-category $\cat{MULTICAT}$ is powered.
I am content if you can construct the power multicategory~$\cat{E} \pitchfork \cat{M}$ and the necessary functor $\cat{E} \mto \cat{M}_\cat{MULTICAT}(\cat{E} \pitchfork \cat{M}, \cat{M})$ and show that any multicategory~$\cat{D}$ with a functor~$\cat{E} \mto \cat{M}_\cat{MULTICAT}(\cat{D}, \cat{M})$ has a 1-cell from $\cat{D}$ to $\cat{E} \pitchfork \cat{M}$.
The remaining requirements need to hold but do not need to be proved.
\end{exercise}

\begin{proof}
Let $\cat{C}_\cat{M}$ be the category whose objects are the same as $\cat{M}$ and whose morphisms are the unary morphisms of $\cat{M}$ with the obvious identity and composition.
Suppose $F_1, \dots, F_n$ and $G$ are functors from $\cat{E}$ to $\cat{C}_\cat{M}$, then let a multitransformation $\alpha$ from $\vec{F}$ to $G$ map an object $\ob{E}$ of $\cat{E}$ to morphism of $\cat{M}$ from $[F_1(\ob{E}), \dots, F_n(\ob{E})]$ to $G(\ob{E})$, and let $\alpha$ be natural if for every morphism $\mo{e} : \ob{E} \mto \ob{E}'$ of $\cat{E}$ the composition $\alpha_\ob{E} \cocomp G(\mo{e})$ equals the composition $[F_1(\mo{e}), \dots, F_n(\mo{e})] \cocomp \alpha_{\ob{E}'}$.

Define $\cat{E} \pitchfork \cat{M}$ to be the category whose objects are functors from $\cat{E}$ to $\cat{C}_\cat{M}$ and whose morphisms are natural multitransformations with composition and identity each defined pointwise (which obviously always results in a natural multitransformation).
Given an object~$\ob{E}$ of $\cat{E}$, let $\pi_\ob{E} : \cat{E} \pitchfork \cat{M} \mto \cat{M}$ map the functor $F : \cat{E} \mto \cat{C}_\cat{M}$ to the object $F(\ob{E})$ and the multitransformation $\alpha$ to the morphism $\alpha_\ob{E}$, which defines a functor of multicategories because composition and identity in $\cat{E} \pitchfork \cat{M}$ are defined pointwise.
Given a morphism~$\mo{e} : \ob{E} \mto \ob{E}'$ of $\cat{E}$, let $\pi_\mo{e} : \pi_\ob{E} \nto \pi_{\ob{E'}}$ be the transformation mapping a functor $F : \cat{E} \mto \cat{C}_\cat{M}$ to the morphism $F(\mo{e}) : [F(\ob{E})] \mto F(\ob{E}')$, which is natural due to the naturality requirement on the morphisms of $\cat{E} \pitchfork \cat{M}$.
$\pi : \cat{E} \mto \cat{M}_\cat{MULTICAT}(\cat{E} \pitchfork \cat{M}, \cat{M})$ is functorial because each object of $\cat{E} \pitchfork \cat{M}$ is functorial.
\end{proof}

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