Note that since \$$\newcommand{\op}[]{^\mathrm{op}}\cat{X}\op\neq\cat{X}\$$, both snake equations must be checked.

The first: \begin{align}\lambda\_\cat{X}\circ(\epsilon\_\cat{X}\times U\_\cat{X})\circ\alpha\_{\cat{X,X\op,X}}^{-1}\circ(U\_\cat{X}\times\eta\_\cat{X})\circ\rho\_\cat{X}^{-1}(x,y)=&\bigvee\_{(z\_1,1)\in\cat{X}\times\textbf{1}}\rho\_\cat{X}^{-1}(x,(z\_1,1))\otimes\lambda\_\cat{X}\circ(\epsilon\_\cat{X}\times U\_\cat{X})\circ\alpha\_{\cat{X,X\op,X}}^{-1}\circ(U\_\cat{X}\times\eta\_\cat{X})((z\_1,1),y)\\\\=&\bigvee\_{(z\_1,1)\in\cat{X}\times\textbf{1}}\cat{X}(x,z\_1)\otimes\lambda\_\cat{X}\circ(\epsilon\_\cat{X}\times U\_\cat{X})\circ\alpha\_{\cat{X,X\op,X}}^{-1}\circ(U\_\cat{X}\times\eta\_\cat{X})((z\_1,1),y)\\\\=&\bigvee\_{(z\_1,1)\in\cat{X}\times\textbf{1}}\cat{X}(x,z\_1)\otimes\bigvee\_{z\_2\times(z\_3\times z\_4)\in\cat{X}\times(\cat{X}\op\times\cat{X})}(U\_\cat{X}\times\eta\_\cat{X})((z\_1,1),z\_2\times(z\_3\times z\_4))\otimes\lambda\_\cat{X}\circ(\epsilon\_\cat{X}\times U\_\cat{X})\circ\alpha\_{\cat{X,X\op,X}}^{-1}(z\_2\times(z\_3\times z\_4),y)\\\\=&\bigvee\_{(z\_1,1)\in\cat{X}\times\textbf{1}}\cat{X}(x,z\_1)\otimes\bigvee\_{z\_2\times(z\_3\times z\_4)\in\cat{X}\times(\cat{X}\op\times\cat{X})}U\_\cat{X}(z\_1,z\_2)\otimes\eta\_\cat{X}(1,z\_3,z\_4)\otimes\lambda\_\cat{X}\circ(\epsilon\_\cat{X}\times U\_\cat{X})\circ\alpha\_{\cat{X,X\op,X}}^{-1}(z\_2\times(z\_3\times z\_4),y)\\\\=&\bigvee\_{(z\_1,1)\in\cat{X}\times\textbf{1}}\cat{X}(x,z\_1)\otimes\bigvee\_{z\_2\times(z\_3\times z\_4)\in\cat{X}\times(\cat{X}\op\times\cat{X})}\cat{X}(z\_1,z\_2)\otimes\cat{X}(z\_3,z\_4)\otimes\lambda\_\cat{X}\circ(\epsilon\_\cat{X}\times U\_\cat{X})\circ\alpha\_{\cat{X,X\op,X}}^{-1}(z\_2\times(z\_3\times z\_4),y)\\\\=&\bigvee\_{(z\_1,1)\in\cat{X}\times\textbf{1}}\cat{X}(x,z\_1)\otimes\bigvee\_{z\_2\times(z\_3\times z\_4)\in\cat{X}\times(\cat{X}\op\times\cat{X})}\cat{X}(z\_1,z\_2)\otimes\cat{X}(z\_3,z\_4)\otimes\bigvee\_{(z\_5\times z\_6)\times z\_7\in(\cat{X}\times\cat{X}\op)\times\cat{X}}\alpha\_{\cat{X,X\op,X}}^{-1}(z\_2\times(z\_3\times z\_4),(z\_5\times z\_6)\times z\_7)\otimes\lambda\_\cat{X}\circ(\epsilon\_\cat{X}\times U\_\cat{X})((z\_5\times z\_6)\times z\_7,y)\\\\=&\bigvee\_{(z\_1,1)\in\cat{X}\times\textbf{1}}\cat{X}(x,z\_1)\otimes\bigvee\_{z\_2\times(z\_3\times z\_4)\in\cat{X}\times(\cat{X}\op\times\cat{X})}\cat{X}(z\_1,z\_2)\otimes\cat{X}(z\_3,z\_4)\otimes\bigvee\_{(z\_5\times z\_6)\times z\_7\in(\cat{X}\times\cat{X}\op)\times\cat{X}}\cat{X}(z\_2,z\_5)\otimes\cat{X}\op(z\_3,z\_6)\otimes\cat{X}(z\_4,z\_7)\otimes\lambda\_\cat{X}\circ(\epsilon\_\cat{X}\times U\_\cat{X})((z\_5\times z\_6)\times z\_7,y)\\\\=&\bigvee\_{(z\_1,1)\in\cat{X}\times\textbf{1}}\cat{X}(x,z\_1)\otimes\bigvee\_{z\_2\times(z\_3\times z\_4)\in\cat{X}\times(\cat{X}\op\times\cat{X})}\cat{X}(z\_1,z\_2)\otimes\cat{X}(z\_3,z\_4)\otimes\bigvee\_{(z\_5\times z\_6)\times z\_7\in(\cat{X}\times\cat{X}\op)\times\cat{X}}\cat{X}(z\_2,z\_5)\otimes\cat{X}\op(z\_3,z\_6)\otimes\cat{X}(z\_4,z\_7)\otimes\bigvee\_{(1,z\_8)\in\textbf{1}\times\cat{X}}(\epsilon\_\cat{X}\times U\_\cat{X})((z\_5\times z\_6)\times z\_7,(1,z\_8))\lambda\_\cat{X}((1,z\_8),y)\\\\=&\bigvee\_{(z\_1,1)\in\cat{X}\times\textbf{1}}\cat{X}(x,z\_1)\otimes\bigvee\_{z\_2\times(z\_3\times z\_4)\in\cat{X}\times(\cat{X}\op\times\cat{X})}\cat{X}(z\_1,z\_2)\otimes\cat{X}(z\_3,z\_4)\otimes\bigvee\_{(z\_5\times z\_6)\times z\_7\in(\cat{X}\times\cat{X}\op)\times\cat{X}}\cat{X}(z\_2,z\_5)\otimes\cat{X}\op(z\_3,z\_6)\otimes\cat{X}(z\_4,z\_7)\otimes\bigvee\_{(1,z\_8)\in\textbf{1}\times\cat{X}}\epsilon\_\cat{X}(z\_5,z\_6,1)\otimes U\_\cat{X}(z\_7,z\_8)\otimes\lambda\_\cat{X}((1,z\_8),y)\\\\=&\bigvee\_{(z\_1,1)\in\cat{X}\times\textbf{1}}\cat{X}(x,z\_1)\otimes\bigvee\_{z\_2\times(z\_3\times z\_4)\in\cat{X}\times(\cat{X}\op\times\cat{X})}\cat{X}(z\_1,z\_2)\otimes\cat{X}(z\_3,z\_4)\otimes\bigvee\_{(z\_5\times z\_6)\times z\_7\in(\cat{X}\times\cat{X}\op)\times\cat{X}}\cat{X}(z\_2,z\_5)\otimes\cat{X}\op(z\_3,z\_6)\otimes\cat{X}(z\_4,z\_7)\otimes\bigvee\_{(1,z\_8)\in\textbf{1}\times\cat{X}}\cat{X}(z\_5,z\_6)\otimes\cat{X}(z\_7,z\_8)\otimes\cat{X}(z\_8,y)\\\\=&\bigvee\_{z\_1,\ldots{},z\_8\in\cat{X}}\cat{X}(x,z\_1)\otimes\cat{X}(z\_1,z\_2)\otimes\cat{X}(z\_3,z\_4)\otimes\cat{X}(z\_2,z\_5)\otimes\cat{X}\op(z\_3,z\_6)\otimes\cat{X}(z\_4,z\_7)\otimes\cat{X}(z\_5,z\_6)\otimes\cat{X}(z\_7,z\_8)\otimes\cat{X}(z\_8,y)\\\\=&\bigvee\_{z\_1,\ldots{},z\_8\in\cat{X}}\cat{X}(x,z\_1)\otimes\cat{X}(z\_1,z\_2)\otimes\cat{X}(z\_2,z\_5)\otimes\cat{X}(z\_5,z\_6)\otimes\cat{X}\op(z\_3,z\_6)\otimes\cat{X}(z\_3,z\_4)\otimes\cat{X}(z\_4,z\_7)\otimes\cat{X}(z\_7,z\_8)\otimes\cat{X}(z\_8,y)\\\\=&\bigvee\_{z\_3,z\_6\in\cat{X}}\cat{X}(x,z\_6)\otimes\cat{X}\op(z\_3,z\_6)\otimes\cat{X}(z\_3,y)\\\\=&\bigvee\_{z\_3,z\_6\in\cat{X}}\cat{X}(x,z\_6)\otimes\cat{X}(z\_6,z\_3)\otimes\cat{X}(z\_3,y)\\\\=&\cat{X}(x,y)\\\\=&U\_\cat{X}(x,y)\end{align}. So, the first snake equation holds.