So in a proof of a theorem in Kanamori's book, we first assume that $\forall a \in {^\omega\omega}(a^\# \text{ exists})$ and then we have two reals $a, b$ and we code them through $c$ and we want to work with $L[a]$ and $L[b]$ in $L[c^\#]$. So naturally I wanted to show that $c \in L[c^\#]$ and go about my way. But here I come to a problem. Let me give some context first.
First there is this theorem:
Theorem. $0^\#$ is absolute for transitive $\in$-models $M$ of ZF such that $\omega_1 \subset M$ in the following sense: $$M \models \ulcorner \text{There is an EM blueprint satisfying (I)-(III)}\urcorner \text{ iff } 0^\# \in M, $$ in which case $M \models \ulcorner 0^\# \text{ is the unique EM blueprint satisfying (I)-(III)} \urcorner$.
[Here the conditions $\text{(I)-(III)}$ are the wellfoundedness, unboundedness and remarkability properties of $0^\#$]
So this theorem is proved by constructing a $\Pi^1_2$ formula which defines the singleton $\{0^\#\}$ and by shoenfield absoluteness this proves the above theorem. For $a \in {^\omega\omega}$, Kanamori says that the exact same procedure above, gives us the absoluteness of $a^\#$ for models that contain it.
But there is a small twist to it that he doesn't mention. When I tried doing this myself to verify his claim, I realized that the above $M$ must also contain $a$ and at the time I didn't think much of it since it didn't matter. But now I am dealing with $L[c^\#]$ and I need to either refine the version of the theorem which I tried proving which required both of $c$ and $c^\#$ to be in $M$. Or more directly, show that we can construct $c$ from $c^\#$ in an explicit way.
So my question ultimately boils down to: How can we show that $c \in L[c^\#]$?
Because of the inherent complexity of this subject to a beginner like me, I would really appreciate any hints or remarks.
Edit I:
To clarify, the reason for the fact that removing the base assumption $c \in M$, from the theorem would work is that then by $c^\# \in M$ we could trivially get to the fact that the EM blueprint for $L[c]$ exists and that would imply that $c \in M$.
You have to be careful about the structure in question! A sharp of a set $a$ doesn't just code the theory of $L[a]$ (plus indiscernible), but rather the expansion of this structure by a constant symbol naming $a$ (look carefully at the definition on page $110$).
So the real in question is built right into the structure whose blueprint our sharp is coding: our theory talks directly about it.