Representable Functors

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Exercises

1. Show that the hom-functors map identity morphisms in $\mathcal{C}$ to corresponding identity functions in $\mathbf{Set}$.
2. Show that Maybe is not representable.
3. Is the Reader functor representable?
4. Using Stream representation, memoize a function that squares its argument.
5. Show that tabulate and index for Stream are indeed the inverse of each other. (Hint: use induction.)
6. The functor:

Pair a = Pair a a

is representable. Can you guess the type that represents it? Implement tabulate and index.