Composite of Surjection on Injection is not necessarily Either

Theorem
Let $f$ be an injection.

Let $g$ be a surjection.

Let $g \circ f$ denote the composition of $g$ with $f$.

Then it is not necessarily the case that $g \circ f$ is either a surjection or an injection.

Proof
Let $X, Y, Z$ be sets defined as:

Let $f: X \to Y$ be defined in two-row notation as:


 * $\dbinom {a \ b \ c } {1 \ 2 \ 3}$

which is seen by inspection to be an injection.

Let $g: Y \to Z$ be defined in two-row notation as:


 * $\dbinom {1 \ 2 \ 3 \ 4} {x \ y \ y \ z}$

which is seen by inspection to be a surjection.

The composition $f \circ g$ is seen to be:


 * $\dbinom {a \ b \ c} {x \ y \ y}$

which is:
 * not an injection (both $b$ and $c$ map to $y$)
 * not a surjection (nothing maps to $z$).

Hence the result.