Notation in Group cohomology and cochains Group Cohomology: Question $1$
I am learning group cohomology. In the Wikipedia, I couldn't understand few terminologies.
For example, in the section $$\text{The functors $\text{Ext}^n$ and formal definition of group cohomology}, \ \cdots \cdots (*)$$
$d:C^n \to C^{n+1}$ is the couboundary operator of non-homogeneous cochains $\varphi_n: G^n \to M$ while $\delta:C^n \to C^{n+1}$  is the coboundary operator on homogeneous cochains.
The general definition of $d$ is given in the previous section.
But what is the general definition of $\delta$ ?
In the above section $(*)$ there is the following para:
$\text{This construction initially leads to a coboundary operator that acts on the "homogeneous" cochains. These are the}$ elements of $\text{Hom}_G(F, M)$, that is, functions $\color{red}{\varphi_n}:G^n \to M$ that obey
$$g \phi_n(g_1,g_2, \cdots, g_n)=\phi_n(gg_1, gg_2, \cdots, gg_n).$$
What is $\phi_n$ here ?
I think there is a typo and the red color $\color{red}{\varphi_n}$ should be $\phi_n$.
Finally, it defines $\varphi_2(g_1,g_2)=\phi_3(1,g_1,g_1g_2)$ and so on. How it follows ?
Any comment please.
 A: As stated in wikipedia, the general definition for inhomogeneous cochains of the homomorphisms
$d =d_n: C^n(G,M)\rightarrow C^{n+1}(G,M)$ is as follows.
If $f\in C^n(G,M)$ then define $d_n(f)$ by
\begin{align*}
d_n(f)(x_1,\ldots , x_{n+1}) & = x_1f(x_2,\ldots , x_{n+1})\\
 & + \sum_{i=1}^n (-1)^i f(x_1,\ldots , x_{i-1},x_ix_{i+1}, \ldots , x_{n+1}) \\
 & + (-1)^{n+1} f(x_1,\ldots,x_n)
\end{align*}
I write $\color{red}{f}$ instead of $\varphi$ for a better distinction between $f$ and the other $\phi$.
For $n=0,1,2,3$ we obtain
\begin{align}
(d_0f)(x_1) & = x_1f-f \\
(d_1f)(x_1,x_2) & = x_1f(x_2)-f(x_1x_2)+f(x_1)\\
(d_2f)(x_1,x_2,x_3) & = x_1f(x_2,x_3)-f(x_1x_2,x_3)+f(x_1,x_2x_3)-f(x_1,x_2)
\end{align}
\begin{align}
\begin{split}
(d_3f)(x_1,x_2,x_3,x_4) & = x_1f(x_2,x_3,x_4)-f(x_1x_2,x_3,x_4)+f(x_1,x_2x_3,x_4)\\
 & -f(x_1,x_2,x_3x_4)+f(x_1,x_2,x_3)
\end{split}
\end{align}
Now the homogeneous variant is defined recursively by the inhomogeneous invariant. In wikipedia it says
"The relation to the coboundary operator $d$ that was defined in the previous section, and which acts on the "inhomogeneous" cochains
$f$, is given by reparameterizing so that...."
And then an example computation is given for $(d_2f)(g_1,g_2,g_3)$. The result is exactly the formula from above for $n=2$ when defining $\phi_3(g_1,g_2,g_3):=f_2(g_1,g_2)$.
This generalises by interating, i.e., define the general formula for $\phi_n$ recursively this way in terms of the $(d_if)$, which are given above.
Actually, it is all explained at wikipedia, but the confusion between $\varphi_n$ and $\phi_n$ is a little bit disturbing.
