Schröder-Bernstein theorem and injective maps

I am a bit confused about the statement of the Schröder-Bernstein theorem which states the following:

Suppose that $A$ and $B$ are sets, and that $f : A to B$ and $g : B to A$ are injective mappings. Then
there exists a bijection $h : A to B$.

• Can it be directly infered that the $f$ and $g$ must only be bijective otherwise that would contradict that both $A$ and $B$ are the same size easily infered from $f$ and $g$?
• And if so, why then the statement of theorem describes $f$ and $g$ as “injective” rather than “bijective”?