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\homework{Categories}

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\begin{exercise}
Give, for any category~$\cat{C}$ and any object~$\ob{C} : \cat{C}$, a monoidal structure on the set~$\ob{C} \mto \ob{C}$.
\end{exercise}

\begin{exercise}
Prove that for any monoid~$\alg{M}$ there is a category with one object~$\star$ such that $\star \mto \star$ equals~$M$.
\end{exercise}

\begin{exercise}
Show that the above extends to a functor from~$\cat{Mon}$ to~$\cat{Cat}$.
\end{exercise}

\begin{exercise}
Show that there is a functor~$F : \cat{Set} \mto \cat{Mon}$ and a functor~$U : \cat{Mon} \mto \cat{Set}$ such that $F \cocomp U$ equals $\List$. Hint: $U$~maps a monoid to its underlying set.
\end{exercise}

\begin{exercise}
Prove that any category that has exactly one functor to it from each other category must be isomorphic to the category~$\cat{1}$.
\end{exercise}

\begin{exercise}
Prove that any category that has exactly one functor from it to each other category must be isomorphic to the category~$\cat{0}$.
\end{exercise}

\begin{exercise}
Given categores $\cat{A}$ and~$\cat{B}$, define a category~$\cat{A} \times \cat{B}$ with ``projection'' functors $\pi_A$ and~$\pi_B$ from it to $\cat{A}$ and~$\cat{B}$ respectively.
\end{exercise}

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