For $\phi \in C^1(X; G)$ a cocycle being thought of as a function from paths in X to G, I want to show: $\phi(f \cdot g) = \phi(f) \cdot \phi(g)$.
What I'm not sure is how I'm supposed to relate a path to a singular simplex? Once I'm able to do this the rest of the problem should just be playing around with the fact that $\delta\phi = \phi\delta = 0$.
On another note shouldn't this just be trivially true since $\phi$ is a homomorphism? Why should it matter if $\phi$ is a cocycle or not?