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if you are still uncomfortable with the definition of composition in the clys Lee category we will show in this section that the classic category is equivalent to the full subcategory of free objects in the island burg more category recall that the clys Lee category is the initial object in the category of T inducing a joint situations so given in a joint situation FG there is a unique functor J from the clys Lee category to a such that j ft is equal to f and g j is equal to u T and recall that Jay takes a morphism F to F F followed by the co unit of FG on component X Prime we will now prove that J is full and faithful for arbitrary objects X and X Prime and E the home set of morphisms in the clys Lee category from X to X prime is by definition the set of morphisms from X to GF x prime in E since t is equal to g f then by the injunction this is isomorphic to the home set and a from FX to FX prime we have to verify that this isomorphism is given by J acting on morphisms we have f as a