The copnumber for lexicographic products and sums of graphs

For the lexicographic product $G\bullet H$ of two graphs $G$ and $H$ so that $G$ is connected, we prove that if the copnumber $c(G)$ of $G$ is greater than or equal to $2$, then $c(G\bullet H)=c(G)$. Moreover, if $c(G)=c(H)=1$, then $c(G\bullet H)=1$. If $c(G)=1$, $G$ has more than one vertex, and $c(H)\geq 2$, then $c(G\bullet H)=2$. We also provide the copnumber for general lexicographic sums.