Elucidation of the role of hydration water underpinning dynamical crossover in proteins has proven challenging. Indeed, many contradictory findings in the literature seek to establish either causal or correlative links between water and protein behavior. Here, via molecular dynamics, we compute the temperature dependence of mean-square displacement and translational self-diffusivities for both hen egg white lysozyme and its hydration layer from 190 to 300 K. We find that the protein's mobility increases sharply at similar to 230 K, indicating dynamical onset; concerted motion with hydration-water molecules is evident up to similar to 285 K, confirming dynamical correlation between them. Exploring underlying mechanisms of such concerted motion, we scrutinize the water protein hydrogen-bonding network as a function of temperature, noting sharp deviation from linearity of the hydrogen bond number's profile with temperature originating near the protein dynamical transition. Our studies reveal a common temperature profile/dependence of self-diffusivity values of the protein, hydration water, and the bulk solvent, originating from a common dependence on the bulk solvent viscosity, eta(s). The key mechanistic role adopted by the protein water hydrogen bond network in relation to the onset of proteins' dynamical transition is also discussed.