  Let ${\mathcal G}$ be a graph family defined on a common (labeled) vertex set
$V$. A set $S\subseteq V$ is said to be a simultaneous metric generator for
${\cal G}$ if for every $G\in {\cal G}$ and every pair of different vertices
$u,v\in V$ there exists $s\in S$ such that $d_{G}(s,u)\ne d_{G}(s,v)$, where
$d_{G}$ denotes the geodesic distance. A simultaneous adjacency generator for
${\cal G}$ is a simultaneous metric generator under the metric
$d_{G,2}(x,y)=\min\{d_{G}(x,y),2\}$. A minimum cardinality simultaneous metric
(adjacency) generator for ${\cal G}$ is a simultaneous metric (adjacency)
basis, and its cardinality the simultaneous metric (adjacency) dimension of
${\cal G}$. Based on the simultaneous adjacency dimension, we study the
simultaneous metric dimension of families composed by lexicographic product
graphs.
