Thermo-sensitive triple-shape polymers can perform two consecutive shape changes in response to heat. These shape changes correspond to the recovery of two different deformations in reverse order, which were programmed previously at elevated temperature levels (Tmid and Thigh) by the application of external stress. Recently, an AB copolymer network was described, which surprisingly exhibited a triple-shape effect despite being programmed with only one deformation at Thigh. Here it is explored whether a copolymer network system can be designed that enables a one-step deformation process at ambient temperature (cold drawing) as a novel, gentle, and easy-to-handle triple-shape-creation procedure, in addition to the procedures reported to date, which generally involve deformation(s) at elevated temperature(s). A copolymer-network system with two crystallizable polyester segments is synthesized and characterized, fulfilling two crucial criteria. These materials can be deformed at ambient temperature by cold drawing and show, even at Thigh, which is above the melting points of both switching domains, elongation at break of up to 250%. Copolymer networks with PCL contents of 75 and 50 wt% show a triple-shape effect after cold drawing with shape-fixity ratios between 65% and 80% and a total-shape-recovery ratio above 97%. Furthermore, in these copolymer networks, the triple-shape effect can be obtained after a one-step deformation at Thigh. Independent of the temperature at which the deformation is applied (ambient temperature or Thigh), copolymer networks that have the same compositions show similar switching temperatures and proportioning of the recovery in two steps. The two-step programming procedure enables a triple-shape effect in copolymer networks for an even broader range of compositions. This versatile triple-shape-material system based on tailored building blocks is an interesting candidate material for applications in fixation systems or disassembling systems.