Well-definedness of $\exp_\tilde p \circ i\circ \exp_p^{-1}:S^n-\{q\}\rightarrow \tilde M$

Question

I am reading the 4.1 Theorem of do Carmo's Riemannian Geometry (as pictures below). The well-definedness of red line is not obvious for me.
In fact, I can accept that $\exp_p^{-1}$ map $S^n-\{q\}$ to $B(0,\pi)\subset T_pM$, where $B(0,\pi)$ is open ball of $T_pM$. Besides, $i$ is ok. But $\exp_{\tilde p}$ is not clear. So, I want to know why the red line is well-defined.