  Let $(X, d, \mu)$ be a metric measure space, with $\mu$ a Borel regular
measure. In this paper, we prove that, if $u\in
L^1_{{\mathop\mathrm{\,loc\,}}}(X)$ and $g$ is a Haj{\l}asz gradient of $u$,
then there exists $\widetilde u$ such that $\widetilde u=u$ almost everywhere
and $4g$ is a $p$-weak upper gradient of $\widetilde u$. This result avoids a
priori assumption on the quasi-continuity of $u$ used in [Rev. Mat.
Iberoamericana 16 (2000), 243-279]. As an application, an embedding of the
Morrey-type function spaces based on Haj{\l}asz-gradients into the
corresponding function spaces based on upper gradients is obtained. We also
introduce the notion of local Haj{\l}asz gradient, and investigate the
relations between local Haj{\l}asz gradient and upper gradient.
