I was wondering if anyone could provide me with some information on the following: Suppose <F,G,eta,epsilon>:A -> B is an adjunction and let C denote the collection of all possible finite horizontal compositions of F,G (as identity natural transformations),eta, and epsilon. So, for example the horizontal composite epsilon.F.eta is a natural transformation in C. I am interested in knowing more about vertical compositions using members of C. For example the vertical composite of epsilon.F with F.eta is epsilon.F.eta, and the vertical composite of epsilon.F.G.F with epsilon.F.eta.eta is epsilon.epsilon.F.eta.eta. Must C be closed under all possible vertical compositions? Any assistance or reference would be appreciated. Sincerely, Andrew Ensor.