We construct a universal code for stationary and memoryless classical-quantum channel as a quantum version of the universal coding by Csisz\\'{a}r and K\\ {o}rner. Our code is constructed by the combination of irreducible representation, the decoder introduced through quantum information spectrum, and the packing lemma. Also, we have proven that there exists a quantum state approximating any multi-copy state universally when we measure the error by means of the normalized relative entropy. While the qubit case was proven by Krattenthaler and Slater, the general case has been open for more than ten years. For a deeper analysis, we have solved the mini-max problem concerning `approximation error' up to the second order. Furthermore, we have applied this result to quantum lossless data compression, and have constructed a universal quantum lossless data compression.
