Qn 4. (a)(i) Define what is meant by \{ A \stackrel{u_i}{\longrightarrow} B_i \}_{i \in I} is a monomorphic family of arrows in a category A.
(ii) Give a specific example to show that the members of a monomorphic family, in a given category, need not be monomorphisms. [3, 8]
(b) Let P: Set \longrightarrow Set be the contravariant power set functor. Then prove the following:
(i) P is faithful but not full.
(ii) P reflects isomorphisms. [8, 7]
(c) Demonstrate that in the category Cat of small categories, monomorphisms are faithful functors. [10]