I am following Cohomological Methods in group theory by Ararat Babakhanian.
Before define cup product, he proves a theorem saying existence of a function.
Let $(X_*,\partial)$ be a complete resolution of $G$. There exists $G$ homomorphisms $\varphi:X_{p+q}\rightarrow X_p\otimes X_q$ for every $p,q\in \mathbb{Z}$ such that $$\varphi_{p,q} \partial =\partial' \varphi_{p+1,q}+(-1)^p\partial'' \varphi_{p,q+1}\\ (\epsilon\otimes\epsilon) \varphi_{0,0}=\epsilon.$$ Where $\partial'=\partial\otimes 1$ and $\partial''=1\otimes \partial$
Proof goes like this :
Assume for one $p$ and all $q$, $\varphi _{p,q}$ is defined and $$\partial'\varphi_{p,q}\partial=(-1)^p \partial'\partial''\varphi_{p,q+1}$$
There is a homotopy $D'$ corresponding to $\partial'$ we then have $\partial'D'+D'\partial'=1$
$$\begin{align*} \varphi_{p,q}\partial&=(\partial'D'+D'\partial')\varphi_{p,q}\partial\\ &=\partial'D'\varphi_{p,q}\partial+D'\partial'\varphi_{p,q}\partial\\ &=\partial'D'\varphi_{p,q}\partial+(-1)^p(D' \partial')\partial''\varphi_{p,q+1}\\ &=\partial'D'\varphi_{p,q}\partial+(-1)^p(1-\partial'D')\partial''\varphi_{p,q+1}\\ &=\partial'(D'\varphi_{p,q}\partial+(-1)^{p+1}D'\partial''\varphi_{p,q+1})+(-1)^p\partial''\varphi_{p,q+1}\\ \end{align*}$$
Defining $\varphi_{p+1,q}=D'\varphi_{p,q}\partial+(-1)^{p+1}D'\partial''\varphi_{p,q+1}$ we have
$$\varphi_{p,q} \partial =\partial' \varphi_{p+1,q}+(-1)^p\partial'' \varphi_{p,q+1}$$
To define $\varphi_{p+2,q}$ as above, we need to have $$\partial'\varphi_{p+1,q}\partial=(-1)^{p+1} \partial'\partial''\varphi_{p+1,q+1}$$
This is where I am stuck.
$$\begin{align*} \partial'\varphi_{p+1,q}\partial&=\partial'(D'\varphi_{p,q}\partial+(-1)^{p+1}D'\partial''\varphi_{p,q+1})\partial\\ &=(-1)^{p+1}\partial'D'\partial''\varphi_{p,q+1}\partial\\\end{align*}$$
Tried substituting $\varphi$ but could not succeed.