Let$A$ and $B$ be two sets. When we say there is a bijective correspondence between $A$ and $B$, it means there is a bijective map between them.
In some texts, to prove there is a correspondence between $A$ and $B$, just show that correspondence to every element of $A$ there is an element in $B$ and conversely. While I think we should prove that there is a well-defined surjective map from $A$ onto $B$. Am I right? Please explain it.
It sounds like you're thinking of the Schröder–Bernstein theorem: If there is an injective map from $A$ to $B$, and an injective map from $B$ to $A$, then there exists a bijective correpondence between $A$ and $B$.
It is necessary that the given maps be injective; otherwise, any two non-empty sets would satisfy the conditions, because you could map every element in one set to a single element in the other.
You can think of what's going on in terms of cardinality, or sizes of the sets. If we have an injection $A\to B$, then we know that $|A|\le |B|$. If we also have an injection $B\to A$, then we know that $|B|\le |A|$. Having both of those inequalities implies that $A$ and $B$ actually have the same cardinality. What I've just said is not a proof of Schröder–Bernstein, though. In fact, it's the other way around.
The proof of Schröder–Bernstein is interesting, and not completely trivial. I recommend the link above, for learning more about it.