 Solution

An inner model of ZFC is a transitive class containing every ordinal and satisfying all its axioms. Write this class as $A$ with its inherited membership relation. Satisfaction is interpreted by restricting all quantifiers to $A$. In a first-order formulation this is a schema for a definable class, possibly with fixed parameters. The class may equal the whole universe. Transitivity means $x\in y\in A\Rightarrow x\in A$; containing all ordinals rules out treating an arbitrary transitive set model as an inner model.