Given a semi-simplicial set $X_\bullet$ we can define its cone $CX_\bullet$ at each level by $CX_0=X_0\amalg\{*\} $ and $CX_n=X_n\amalg X_{n-1}$. The simplicial operators send a new element of $CX_n$, e.g. $\sigma'\in X_{n-1} $ to the compatible elements in $X_{n-2}$ and to $\sigma$ itself.
I want to show that it is "contractible", that is quasi-isomorphic, to $C\{\emptyset\}$, and the quasi-isomorphism is given by the inclusion $i:C\{\emptyset\}\to CX$.
Some calculations show that the only nonzero epimorphism $g:CX\to C\{\emptyset\}$ is $g_n=0$ for $n>0$ and $g_0(\sigma)=*$ for every generator $\sigma$.
Of course, $g\circ i=\text{Id}_{C\{\emptyset\}}$. However, I can't show that $i\circ g$ is homotopic (this is a hint in the problem) to the identity on $CX$. Indeed, I need a function $$K:\mathbb{Z}(X_0)\oplus \mathbb{Z}(*)\to \mathbb{Z}(X_1)\oplus \mathbb{Z}(X_0) $$ satisfying $$ \partial\circ K(a,b)=(0,a+b). $$ Geometrically, I need to find for each vertex in $X$ a combination of edges such that their boundary is $*$. However, the boundary of edges is always "even". So this is where I'm stuck.
Am I doing something wrong? One problem which might be is that I'm trying to find a homotopy for the zero's component. That is $K_0:C_0(CX)\to C_1(CX) $ such that $\partial K_0=f_0\circ g_0$. I mean, perhaps the condition for homotopy is only for higher orders. Unfortulately, no one stressed to us what happens in the zero level at any time in class.