Let $F$ be the free group on the generators $a_1,\ldots,a_n$. Define homomorphisms $\phi_i:F\to F$ by $\phi_i(a_j)=a_j^{1-\delta_{ij}}$, where $\delta_{ij}$ is the Kronecker delta; basically, $\phi_i$ kills the $i$-th generator and fixes the others. Denote $N=\bigcap_{i=1}^n\ker\phi_i$.
I want to identify the subgroup $N$ or at least find its index in $F$. Clearly we have F'\subseteq N, where F' is the commutator subgroup. Is it in fact the case that N=F'?
Edited: it has been pointed out in the comments that I was very hasty in the previous paragraph. The inclusion I stated holds only if $n\leq2$. However, $N$ is not trivial as it contains the element $[\ldots[a_1,a_2],a_3]\ldots]$