Old Mexican, first and last detective (5)
I believe the answer is:
aztec
'old mexican' is the definition.
(I've seen this before)
'first and last detective' is the wordplay.
'first and' becomes 'a' (1st letter of 'and').
'last' becomes 'Z' (Last letter alphabet).
'detective' becomes 'tec' (tec is short for detective).
'a'+'z'+'tec'='AZTEC'
(Other definitions for aztec that I've seen before include "Old Mexican empire" , "Irish lower house" , "nickname of US State" , "Ancient inhabitant of Mexico before Spanish conquest" , "One of Montezuma's people" .)