Here is the gist of some of the exercises. (I have paraphrased and selected.)

Ex 7. Describe the arrow part of the hom functor \$$Hom_C(B,\cdot) \$$ and the naturality condition.

Ex 10. Definition of bifunctor. Show that a bifunctor is a functor in each of its arguments separately. Plus more technical questions relating to bifunctors.

Ex 11. For naturally isomorphic functors F,G, show:

* F is faithful iff G is faithful
* F is full iff G is full
* If F is naturally isomorphic to the identity functor, then it is fully faithful.
* Relationships between fullness and faithfulness under composition.

Ex 17. Let \$$F,G: C \\implies P\$$ be functors from a category C to a preorder P.

* Describe necessary and sufficient conditions under which there is a natural transformation from F to G.
* Prove that if P and Q are preorders, then the functor category \$$Q\$$ is also a preorder.

Ex 18. Verify that the functor category \$$D^C\$$) is a category.

Ex 19. Prove that the functor category \$$D^2\$$ is essentially the category of arrows of D.