I think the relationship is because all organisms are made up of carbon. Therefore organisms consume carbon and also get consumed by organisms made up of carbon. Food webs and energy pyramid and the carbon cycle are closely related since every living organism is made of carbon. For example, when an envelope or a cow is eating grass, it is consuming carbon, and when a person eats the cow, the person consumes carbon, therefore the carbon moves through the food webs and so it does in the energy pyramid.