We simultaneously characterize the temporal and spectral responses of the coherent component in bright twin beams using a spectrally resolved, delay-dependent sum-frequency generation (SFG) measurement. The measured temporal-correlation width is 0.138 ps, close to the transform-limited value of 0.126 ps estimated from the measured twin-beam spectrum.The observed spectral-correlation width is 0.176 rad/ps and is limited by the spectrometer resolution. The corresponding time-frequency width product is 0.0243, far below the value of 1 for SFG between two independent transform-limited Gaussian pulses.