I have been working my way through Lawvere and Schanuel (1997) without too much trouble, but now that I am up to Article V, I am stumped. So, without further ado:
Exercise 6: In a category with products in which map objects exist for any two objects, there is for any three objects a standard map: $B^A \times C^B \stackrel{\gamma}{\longrightarrow} C^A$ which represents composition in the sense that $\gamma\langle\lceil f \rceil, \lceil g \rceil\rangle = \lceil gf \rceil $ for any $A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C.$
I understand how an exponential map object is like a product map, and I can diagram product maps just fine. They're pretty straightfoward. But for this problem, I think the key to solving the problem lies in isolating each of the $\lceil f \rceil, \lceil g \rceil$ terms in order to calculate the product $\lceil gf \rceil$. In the book, they only give them as parameters of the functions f and g taking A to B and B to C, respectively.
When I was first stuck, I looked up the definition of a exponential map object in wikipedia, but that didn't help me in the calculation. I can write the diagram from A to B and from B to C separately, but I am stuck at the point of putting them together into a composite function.