Let \\(\newcommand{\cat}[1]{\mathcal{#1}}p\in\\cat{P},\,s\in\cat{S}\\). \begin{align}(\Phi\circ\Psi)\circ\Upsilon(p,s)=&\bigvee\_{r\in\cat{R}}(\Phi\circ\Psi)(p,r)\otimes\Upsilon(r,s)\\\\=&\bigvee\_{r\in\cat{R}}\left(\bigvee\_{q\in\cat{Q}}\Phi(p,q)\otimes\Psi(q,r)\right)\otimes\Upsilon(r,s)\\\\\cong&\bigvee\_{r\in\cat{R},q\in\cat{Q}}(\Phi(p,q)\otimes\Psi(q,r))\otimes\Upsilon(r,s)\\\\\cong&\bigvee\_{r\in\cat{R},q\in\cat{Q}}\Phi(p,q)\otimes(\Psi(q,r)\otimes\Upsilon(r,s))\\\\\cong&\bigvee\_{q\in\cat{Q}}\Phi(p,q)\otimes\left(\bigvee\_{r\in\cat{R}}\Psi(q,r)\otimes\Upsilon(r,s)\right)\\\\=&\bigvee\_{q\in\cat{Q}}\Phi(p,q)\otimes(\Psi\circ\Upsilon)(q,s)\\\\=&(\Phi\circ(\Psi\circ\Upsilon))(p,s)\end{align}.

Since \\(\mathcal{V}\\) is skeletal, the isomorphisms in the above calculations are equalities. So the lemma is proved.