Assume that $\phi_i: (X, x_0) \rightarrow (Y, y_0)$, for $i=0,1$ are freely homotopic. Prove that $\phi_{0*}$ and $\phi_{1*}$ are conjugate, meaning :
There is $[\lambda] \in \pi_1(Y,y_0)$ with $\phi_{0*}([f]) = [\lambda]\phi_{1*}([f])[\lambda]^{-1}$ for every $[f]$ in $\pi_1(X,x_0)$.
Is this saying a particular $[\lambda]$ for each different $[f]$? Or is it saying a specific $[\lambda]$ that works for every single $[f]$?
No, this is the same $[\lambda]$ for each $[f]$. This is really saying: $$\exists [\lambda] \in \pi_1(Y,y_0) \text{ s.t. } \forall [f] \in \pi_1(X,x_0), \phi_{0*}[f] = [\lambda] \phi_{1*}[f] [\lambda]^{-1}.$$
Try to prove it and you'll see why.