The average coflow completion time (CCT) is the standard performance metric in coflow scheduling. However, standard CCT minimization may introduce unfairness between the data transfer phase of different computing jobs. Thus, while progress guarantees have been introduced in the literature to mitigate this fairness issue, the trade-off between fairness and efficiency of data transfer is hard to control. This paper introduces a fairness framework for coflow scheduling based on the concept of slowdown, i.e., the performance loss of a coflow compared to isolation. By controlling the slowdown it is possible to enforce a target coflow progress while minimizing the average CCT. In the proposed framework, the minimum slowdown for a batch of coflows can be determined in polynomial time. By showing the equivalence with Gaussian elimination, slowdown constraints are introduced into primal-dual iterations of the CoFair algorithm. The algorithm extends the class of the$\sigma$-order schedulers to solve the fair coflow scheduling problem in polynomial time. It provides a 4-approximation of the average CCT w.r.t. an optimal scheduler. Extensive numerical results demonstrate that this approach can trade off average CCT for slowdown more efficiently than existing state of the art schedulers.