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We investigate the formation, morphology, and morphological evolution of galaxies using the Horizon Run 5 cosmological simulation. We quantify the asymmetry and morphology of the stellar mass component of galaxies with stellar mass M> 2×10^9 M⊙, and classify them into disk, spheroid, and irregular types. We find that the initial morphology of the galaxies in the cosmic morning is dominated by disk types with Sersic index < 1.5. The fraction of disk galaxies is ~2/3, while that of irregular and spheroid galaxies is ~1/6 each. Irregular or spheroidal morphologies are found to be incidental and transient. The fractions are roughly independent of redshift and stellar mass up to 10^10 M⊙. For more massive galaxies, the fraction of spheroidal and irregular types increases, whereas at fixed stellar mass, the disk fraction increases monotonically with time. Using nearly the same morphological parameters, we classify ~19,000 high-redshift galaxies observed by JWST and confirm the dominance of disk galaxies at redshift > 1 and up to 6 with very good quantitative agreement in morphology fractions, their redshift evolution, and stellar mass dependence. We also review additional results from the HR5 simulation.