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