Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be two functions. Then $g \circ f$ is clearly defined $\forall a \in A$ but what about $f \circ g$, do we take it as undefined, given that A and C are disjoint? What if A and C are not disjoint?
Since domain of $g$ is the same as the codomain of $f$ hence $g \circ f$ exists $\forall a \in A$.
What I think: If A and C are disjoint sets, then $f(c)$ is not defined, since $c\in C$ is not in the domain of $f$
Am I missing something?
There is a strict definition for composition:
If $f: A \to B$, $g: C \to D$ are functions, the composition $f \circ g$ is the function $C \to B$ given by $c \mapsto g(c) \mapsto f(g(c))$, which is well defined if and only if $g(c) \in A$.
So indeed your intuition is right.