Does anyone know of a reference for the following fact?

Let $M_g$ denote the moduli stack of genus g curves, let $A_g$ denote the moduli stack of abelian varieties, and let $U_g \rightarrow A_g$ denote the universal abelian variety. For any basepoint $[C] \in M_g$, there is a representation $\pi_1(M_g, [C]) \rightarrow Sp_{2g}(\mathbb Z)$ by sending a loop to a homeomorphism of the curve C. On the other hand, if one applies the torelli map, sending a curve to its Jacobian, $\tau: M_g \rightarrow A_g$, one can consider the Galois representation $\rho_{U_g}: \pi_1^{et}(A_g, \overline{[\tau(C)]}) \rightarrow Sp_{2g}(\widehat{\mathbb Z})$ (the monodromy of the moduli stack of abelian varieties, given by action on the $\ell$ torsion).

Why do these two representations agree? That is, why does the square, with vertical maps given by profinite completion, commute?

$\require{AMScd}$ \begin{CD} \pi_1(M_g, [C]) @>>> Sp_{2g}(\mathbb Z) \\ @VVV @VVV \\ \pi_1^{et}(A_g, \overline{[\tau(C)]}) @>>> Sp_{2g}(\widehat{\mathbb Z}) \end{CD}