Suppose $D$ is an $\infty$-category, then we have the equivalence $$ \text{Fib} (D) \substack{ \text{St} \\ \longrightarrow \\ \cong \\ \longleftarrow \\ \text{Un}} [ D^\text{op}, \mathbf{Kan}]$$ between (right) fibrations over $D$ and functors from the opposite of $D$ to Kan complexes ($\infty$-groupoids) via the Grothendieck construction.

Given a map $f: C \to D$ of $\infty$-categories, there is a functor $f^* : \text{Fib} (D) \to \text{Fib}(C)$ by forming pullbacks. Since $f$ induces a functor $C^\text{opp} \to D^\text{opp}$, there is also a functor $f^* : [ D^\text{op}, \mathbf{Kan}] \to [ C^\text{op}, \mathbf{Kan}]$.

Question: For a fibration $\pi : X \to D$ do we have $$ \text{St} f^* \pi \cong f^* \text{St} \pi \quad ?$$ So is the Grothendieck construction compatible with pullbacks?

Thanks for any hints.