DP coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvo\v{r}\'{a}k and Postle in 2015. The same DP chromatic number $\chi_{DP}$. $\chi_{DP}(G \square H) \leq \text{min}\{\chi_{DP}(G) + \text{col}(H), \chi_{DP}(H) + \text {col}(G) \} – 1$ where $\text{col}(H)$ is the number of colors in graph $H$. We focus on building tools for the lower bound arguments of $\chi_{DP}(G \square H)$ and use them to demonstrate the sharpness of the upper bound and its various forms. Our results show that his DP color function of G, the DP analogue of the chromatic polynomial, is essential for the study of the DP chromatic number of the Cartesian product of graphs. A classical result on the gap between list and chromatic numbers: Given any graph $G$ and $k \in \mathbb{N}$, $\chi_{DP}(G \square K_ {k,t})= \chi_{DP}(G) + k$?
