The diffeomorphisms induce actions or pullbacks, formula 119. This shows that pullbacks exist in any category with binary products and equalizers. One can therefore talk about lifting products, equalizers, pullbacks, etc. Unlike, say, trajectories in formula 55, pullbacks are linear objects even for nonlinear formula 55. For example, one can say that "G" preserves products, equalizers, pullbacks, etc. This condition is equivalent to asking that the functor preserves pullbacks and the final object. Any category with fibre products (pullbacks) and products has equalizers. Many steps also have single, double, and triple versions, including pullbacks, timesteps, and drawbacks. This first occurred intra-day on February 29, 2012, before further volatile pullbacks. Taking pullbacks induces natural maps formula 5. Liu even earned a nickname "Liu PaoPao" because of pullbacks in two consecutive Olympic Games. An advantage of Haefliger structures over foliations is that they are closed under pullbacks.