If $\mathcal U$ is a not empty relation in a set $X$ and $\mathcal V$ is a not empty relation in a set $Y$ then it is usual to put $$ \mathcal U\ast\mathcal V:=\big\{\big((a,b),(p,q)\big)\in(X\times Y)\times(X\times Y):(a\mathcal U p)\wedge(b\mathcal V q)\big\} $$ Now let's we assume that $\mathcal U$ and $\mathcal V$ are transitive. So if $\big((a,b),(p,q)\big)$ and $\big((p,q),(h,k)\big)$ are in $\mathcal U\ast\mathcal V$ then $(a,p)$ and $(p,h)$ are in $\mathcal U$ whereas $(b,q)$ and $(q,k)$ are in $\mathcal V$ so that by transitivity $(a,h)$ is in $\mathcal U$ and $(b,k)$ in $\mathcal V$ and thus $\big((a,b),(h,k)\big)$ is in $\mathcal U\ast\mathcal V$: we conclude that $\mathcal U\ast\mathcal V$ is transitive.
Now assuming $\mathcal U\ast\mathcal V$ is transitive then I expected that even $\mathcal U$ and $\mathcal V$ were transitive but unfortunately I was not able to prove it so that I thought to put here a specific question where I ask to prove or disprove (with a counterexample) this. So could someone help me, please?