  Let $G=(V(G),E(G))$ be a simple graph. A set $S \subseteq V(G)$ is a strong
edge geodetic set if there exists an assignment of exactly one shortest path
between each pair of vertices from $S$, such that these shortest paths cover
all the edges $E(G)$. The cardinality of a smallest strong edge geodetic set is
the strong edge geodetic number $\text{sge}(G)$ of $G$. In this paper, the
strong edge geodetic problem is studied on the Cartesian product of two paths.
The exact value of the strong edge geodetic number is computed for $P_n \,
\square \, P_2$, $P_n \, \square \, P_3$ and $P_n \, \square \, P_4$. Some
general upper bounds for $\text{sge}(P_n \, \square \, P_m)$ are also proved.
