Category of symmetric monoidal functors out of a symmetric monoidal category with duals form a groupoud

Question

In Lurie's proof of the cobordism hypothesis, he uses the following fact:

Let $C,D$ be a symmetric monoidal categories with duals, and $Z_1,Z_2\colon C\to D$ be symmetric monoidal functors, then any natrual transformation $\alpha\colon Z_1\to Z_2$ is invertible with inverse given by $$ Z_2\left(M\right)\xrightarrow{\sim}Z_2\left(M^*\right)^*\xrightarrow{\alpha_{M*}^*}Z_1\left(M^*\right)^*\xrightarrow{\sim}Z_1\left(M\right) $$ I don't quite see why this is the inverse.

Answer 1

This is Theorem 3.2 in Duals Invert by Franco, Street, Wood.