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\homework{Transformations}

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\begin{remark}
Proofs by contraditiction or law of excluded middle are not permitted.
\end{remark}

\begin{exercise}
Prove that for any category $\cat{C}$ and any object $\ob{C} : \cat{C}$, the category $\cat{Sub}(\ob{C})$ is thin, meaning there is at most one morphism between any two objects.
\end{exercise}

\begin{exercise}
Prove that $\cat{Prost}$ is a reflective subcategory of $\cat{Rel}(2)$ (the category whose objects are sets with a binary relation and whose morphisms are relation-preserving functions).
\end{exercise}

\begin{remark}
To get an early start on Exercise~3 below, look at Exercise~6 in the lecture notes for Nulls.
\end{remark}

\begin{exercise}
Suppose a subcategory $\cat{S} \xmono{I} \cat{C}$ has a mapping from each object $\ob{C} : \cat{C}$ to a reflection arrow $\ob{C} \xmto{\mo{r}_\ob{C}} I(R(\ob{C}))$.
Prove that there is a unique way to extend the function $R$ to a functor from $\cat{C}$ to $\cat{S}$ such that the reflection arrows form a natural transformation $\mo{r} : \cat{C} \nto R \cocomp I$.
\end{exercise}

\begin{exercise}
Prove that the category $\cat{Cat}$ can be enriched in the multicategory $\cat{CAT}$.
\end{exercise}

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