I am attempting the following exercise from chapter 5 of Van den Dries' notes "Introduction to Model-Theoretic Stability". I suspect the exercise shouldn't be too difficult but I've become pretty stuck. Here, $\mathbb{M}$ is a monster model ($\kappa$-saturated and strongly $\kappa$-homogeneous for some $\kappa$) in some many-sorted language $L$.
Let $Y \subseteq \mathbb{M}_y$ with finite $y$ be definable and let $c$ be a finite tuple. Show: $c$ codes $Y$ if and only if for all $f \in Aut(\mathbb{M})$, $f(Y) = Y \iff f(c) = c$.
I can show "If $c$ codes $Y$ then for all $f \in Aut(\mathbb{M})$, $f(Y) = Y \iff f(c) = c$." with no difficulty. For the other direction I am stuck.
Assuming "for all $f \in Aut(\mathbb{M})$, $f(Y) = Y \iff f(c) = c$", I want to produce a formula $\phi(y,z)$ with no parameters such that $Y = \phi(\mathbb{M}_y, c)$ and $c$ is the unique element of our model with this property, but I can't see how to get such a formula from the condition on automorphisms. I can prove that $Y$ is definable from $c$.
Assume that for all $f\in\mathrm{Aut}(\mathbb{M})$, $f(Y) = Y \iff f(c) = c$. You've noted in the question that this implies $Y$ is definable from $c$, so let $\psi(y,z)$ be a formula such that $Y = \psi(\mathbb{M},c)$.
Let $p(z) = \mathrm{tp}(c/\varnothing)$. Then $p(z)\cup p(z') \cup \{z\neq z', \forall y\, \psi(y,z) \leftrightarrow \psi(y,z')\}$ is inconsistent (do you see why?). By compactness there is a formula $\theta(z)\in p(z)$ such that $\theta(z)\land \theta(z') \land (z\neq z') \land (\forall y\, \psi(y,z) \leftrightarrow \psi(y,z'))$ is inconsistent. Let $\varphi(y,z)$ be $\theta(z)\land \psi(y,z)$, and check that $c$ is the unique element of $\mathbb{M}$ such that $Y = \varphi(\mathbb{M},c)$.