The standard definition of function composition is:
Let $f : A \to B$, $g : B \to C$ then there is a composite function $g \circ f : A \to C$, given by $(g \circ f)(a) = g(f(a))$ with $a \in A$
Why could not be given a less restrictive definition like the following?
If $f : A \to B$, $g : D \to C$, $\operatorname{rg}(f) \subseteq D$ then there is a composite function $g \circ f : A \to C$, given by $(g \circ f)(a) = g(f(a))$ with $ a \in A$
The standard definition (say from Wolfram) is in fact the second one. Also, the standard definition allows the range of the function to be a subset of the stated codomain. Note that if you want the composition to be surjective onto the codomain of the second function, then the range of the first must be the domain of the second, and the range of the second must be its codomain.