Mapper is an algorithm for describing high-dimensional datasets in terms of simple geometric objects. We give a new definition of Mapper, with which we are able to prove that Mapper is a functor and that Mapper is a homotopy equivalence for certain "nice" input data. To establish these results we describe the statistical theory of functorial clustering and the topological machinery of homotopy colimits. At the end of the document we show, by means of numerical experiments, that the functoriality of Mapper is useful in applications.