Given a composition of Bely\u{ \i} maps $\beta \circ \gamma: X \rightarrow Z$, paths between edges of $\beta$ are extended to form loops, then lifted by $\gamma$. These liftings are then studied to understand how loops in $Z$ act on edges of $\beta \circ \gamma$, demonstrating the group operation in $\mathop{ \mathrm{ Mon}} \beta \circ \gamma \unlhd \mathop{ \mathrm{ Mon}} \gamma \wr \mathop{ \mathrm{ Mon}} \beta$. Abstracting away the specific Bely\u{ \i} map $\gamma$ and finding the image of $\pi_{ 1}(Z)$ in $\pi_{ 1}(Y) \wr \mathop{ \mathrm{ Mon}} \beta$ instead allows subsequently determining $\mathop{ \mathrm{ Mon}} \beta \circ \gamma$, for any $\gamma$, using only the monodromy representation of $\gamma$.
Categories: math.NT