  DP-coloring (also called correspondence coloring) is a generalization of list
coloring introduced by Dvo\v{r}\'{a}k and Postle in 2015. Motivated by results
related to list coloring Cartesian products of graphs, we initiate the study of
the DP-chromatic number, $\chi_{DP}$, of the same. We show that $\chi_{DP}(G
\square H) \leq \text{min}\{\chi_{DP}(G) + \text{col}(H), \chi_{DP}(H) +
\text{col}(G) \} - 1$ where $\text{col}(H)$ is the coloring number of the graph
$H$. We focus on building tools for lower bound arguments for $\chi_{DP}(G
\square H)$ and use them to show the sharpness of the bound above and its
various forms. Our results illustrate that the DP color function of $G$, the DP
analogue of the chromatic polynomial, is essential in the study of the
DP-chromatic number of the Cartesian product of graphs, including the following
question that extends the sharpness problem above and the classical result on
gap between list chromatic number and chromatic number: given any graph $G$ and
$k \in \mathbb{N}$, what is the smallest $t$ for which $\chi_{DP}(G \square
K_{k,t})= \chi_{DP}(G) + k$?
