omega-completions

I recently noticed the following fact, and wondered if this was known, and if so, whether there was a reference. Write w for the first infinite ordinal (pronounced omega, and thought of as a category). The free completion under w-colimits of a category K can be described as the category of fractions [w,K][S^-1], where [w,K] is the functor category, and S consists of all natural transformations of the form A(phi):A-->Af, where A:w-->K and f:w-->w are functors, and phi:1_w-->f a natural transformation. Steve Lack.