I am reading Gelfand-Manin, and am a little confused about their proof that the equivalence relationship between roofs in the localization of a category $B$ at a localizing class of morphisms.
In particular, in proving transitivity, it seems that one can just put a roof over the bottom diagram on page 149, and then check all of the commutativity stuff (which I did, I think). But their argument is more complicated than this - in particular, it uses the third axiom of a localizing system ($ft = gt$ is equivalent to $sf = gs$ for some s). What am I missing?
(I'm not sure how to tex up these diagrams, sorry!)