This shows two latin squares $L$ and $M$ of size $4$. We claim that they are orthogonal. To check this, we need to merge them together to form $L*M$.

For $L$ and $M$ to be orthogonal, every possible combination $ij$ (with $i,j\in\{1,2,3,4\}$) must appear precisely once in $L*M$. All possible combinations are laid out in a regular way in the small square on the right. We can click on them to see how they appear in $L*M$.

The combination 11 occurs precisely once in $L*M$, as expected.