Countifs with conditions based on same row in source

I have a large data set where one requirement is to obtain the count of duplicate people using their first and last name provided in two discrete columns and perform a lookup in another sheet for a count of that sheets rows that have the same two values in two different rows.

For example, in the attached document, A2:B13 represents the lookup data I want to check for appearances of rows in E2:F10.
A matching row must have both the first and last name equal, then I want the count in column next to E2:F10.

I know I could create a computed column in each concatenating the first and last name of each sheet to simplify this, but in this case it would more efficient to create a single formula and auto fill it throughout a column.

Is this possible?