In this technical note, the relation between notions of uniform observability of discrete-time nonlinear systems based on injectivity of an observation map (window), the full-rankness of its Jacobian, and kappa-function is investigated. It is proven that a system is uniformly observable in the sense of injectivity of the observation map (together with the full-rankness of its Jacobian) if and only if it is so in some kappa-function senses.