Let A and B be two sets, each with a finite number of elements.

$A$ and $B$ are two non empty sets.

Let $f$ be a function from $A$ to $B$.

It is given that there is injective map from $A$ to $B$.

That means $f$ is one-one function.

It is also given that there is injective map from $B$ to $A$.

That means every element of set $B$ has its image in set $A$.

$\Rightarrow f$ is onto function or surjective.

$\therefore f$ is bijective.

(If a function is both injective and surjective, then the function is bijective.)