For transitive model $M$, is $L^M$ really a model of ZFC? There are two concepts of "models" in set theory: one says that for each axiom $\phi$ of ZFC, $\phi^M$ holds. Remaining one just says that $M\models \mathsf{ZFC}$. First one is a schema rather a single statement. Second one is a single statement and $M$ possibly satisfies the non-standard axioms of ZFC.
We know that $L$ is an inner model of ZFC, that is, every relativized axiom over $L$ holds. My question is, if $M$ is a transitive model of ZFC, then $L^M\models \mathsf{ZFC}$? $L^M$ should be $L_\gamma$ for some $\gamma$, and $L^M$ satisfies all "standard" axioms of ZFC. Can it possible that some non-standard axiom fails on $L^M$? Thanks for any help.
 A: Given an axiom $\varphi$ of $\sf ZFC$, it is provable from $\sf ZF$ that $\varphi^L$ holds. This means that as a meta-theorem, if $M$ is a model of $\sf ZF$, then $L^M$ is a model of $\sf ZFC$.
You are correct to assume that if $M$ is a transitive model, then $L^M$ is $L_\gamma$ for some $\gamma$. Or if $M$ is a class, then $L^M$ is just $L$ again.
But then your question takes a turn into unclear waters. Non-standard axioms occur when you have a non-standard model, with non-standard integers, and internally it has new axioms of $\sf ZFC$ which need not hold externally. If $M$ is transitive, then this is certainly not the case.
Moreover, the truth is a meta-theoretic concept, namely it is not part of $M$ but rather a part of $V$. So asking whether or not a statement is true or false is done in the meta-theory, and non-standard axioms live - by definition - internally to non-standard models, which are models which disagree with the meta-theory by definition.
So in other words, the meta-theory is not aware of any "non-standard axioms", so it certainly cannot tell you whether or not they are true or false in $L^M$, even if $M$ was a non-standard model.
A: The problem I imagine is that formulas are not an object in $V$. As far as I know formulas live outside $V$. As I described it makes some technical problem. 
For example, consider the theory $$ZFC + \exists c:\text{$c$ is countable transitive and $\phi^c$ for each axiom $\phi$ of ZFC}.$$ (It is described in  Ch. 9, section 9 (1b) in Kunen's old set theory. I have seen its name but I forgot it.) Such theory is not $\omega$-consistent and if $V$ is a model of such theory then $V$ should have non-standard naturals. (Is it right?)
Now we gaze the inside of $V$. By Gödel, $c$ does not think that itself satisfies (formalized) ZFC, though it satisfies all "standard" axioms of ZFC.
I think my question is related such situation. When proving $L$ is an inner model of ZFC, we provide $\phi^L$ holds for each axiom $\phi$. But I realized that if we apply same argument to proving $L^M$ is an inner model of $M$ then the argument says that for each (coded) axioms of ZFC $\ulcorner\phi\urcorner$, $M\models \ulcorner\phi^L\urcorner$. By induction for $\ulcorner\phi\urcorner$ we can check that $M\models \ulcorner\phi^L\urcorner$ if and only if $L^M\models \ulcorner\phi\urcorner$. Thus $L^M$ is really a model of ZFC (and in fact a model of $ZFC+ V=L$.) 
Is my reasoning right?
