A. It turns out that the definition of product given above allows a great variety of constructions used in mathematics connected with functions of several variables.

I) Show that if A is also terminal in A then A is isomorphic to I. (ii) Show that if A is isomorphic to I then A is terminal. 2. Consider a product A x B of A and B with projections pi : A x B -+ A, p2 : A x B -+ B. Show that if P is isomorphic to A x B then P with suitable projections is also a product of A and B. 3. Show that if I is a terminal object in a category A and X is any object of A then the product X x I exists, and in fact X is the product of X and I. 4. Consider the two functions f, g:IR ->IR defined by f (t) = cost, g(t) = sin t.

### Categories and Computer Science by R. F. C. Walters

by Jason

4.1