  We consider time correlation for KPZ growth in 1+1 dimensions in a
neighborhood of a characteristics. We prove convergence of the covariance with
droplet, flat and stationary initial profile. In particular, this provides a
rigorous proof of the exact formula of the covariance for the stationary case
obtained in [SIGMA 12 (2016), 074]. Furthermore, we prove the universality of
the first order correction when the two observation times are close and provide
a rigorous bound of the error term. This result holds also for random initial
profiles which are not necessarily stationary.
