I am reading MacLane's Book on Homology and in page $126$ he is talking about obstruction to extensions. My problem is when he says
Suppose now that just the abstract kernel $(C,G,\psi)$ is given.
And now he is trying to see if he can find a function $f$ that is going to satisfies the conditions for a factor set of the extension of a non-Abelian group $G$. One of this conditions in particular is
$[\phi(x)f(y,z)]+f(x,yz)=f(x,y)+f(xy,z)$
He writes that the associative law of $\phi(x)\phi(y)\phi(z)$, where $\phi(x)$ is an element in the automorphism class of $\psi(x)$, shows that the last equation that i wrote holds if $\mu$ is applied to both sides, where $\mu$ gives us the inner automorphisms. I have no idea why this last sentence is true i tried to check it but failed, so any help is appreciated. Thanks in advance!