functions - Correspondence and bijective correspondence between two sets

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.