A system for fastening cooling elements to a component includes a base device with rows of heating elements to support a bottom side of the component, and a gripping device arranged on the base device and including a corresponding number of rows with rail elements configured to temporarily hold the cooling elements on the gripping device and arranged such that an nth row of heating elements is congruent with an nth row of rail elements. An adhesive joint is applied onto a component topside in strips congruent with the rows of heating elements. The gripping device arranges the cooling elements on the strips of the adhesive joint and presses them against the topside of the component, wherein the heating elements heat the adhesive joint between the cooling elements and the component topside to a temperature which is at least as high as a curing temperature of the adhesive joint.