  For a fixed set $X$, an arbitrary \textit{weight structure} $d \in
[0,\infty]^{X \times X}$ can be interpreted as a distance assignment between
pairs of points on $X$. Restrictions (i.e. \textit{metric axioms}) on the
behaviour of any such $d$ naturally arise, such as separation, triangle
inequality and symmetry. We present an order-theoretic investigation of various
collections of weight structures, as naturally occurring subsets of
$[0,\infty]^{X \times X}$ satisfying certain metric axioms. Furthermore, we
exploit the categorical notion of adjunctions when investigating connections
between the above collections of weight structures. As a corollary, we present
several lattice-embeddability theorems on a well-known collection of weight
structures on $X$.
