Let PBm be the category of all principal fibred bundles with m-dimensional bases and their principal bundle homomorphisms covering embeddings. We introduce the concept of the so called (r,m)-systems and describe all gauge bundle functors on PBm of order r by means of the (r,m)-systems. Next we present several interesting examples of fiber product preserving gauge bundle functors on PBm of order r. Finally, we introduce the concept of product preserving (r,m)-systems and describe all fiber product preserving gauge bundle functors on PBm of order r by means of the product preserving (r,m)-systems.