Chapel presents old exotic detective with test (7)
I believe the answer is:
chantry
'chapel' is the definition.
(chantry is a kind of chapel)
'old exotic detective with test' is the wordplay.
'old exotic detective' becomes 'Chan' (Chinese detective Charlie Chan).
'with' says to put letters next to each other.
'test' becomes 'try' (synonyms).
'chan'+'try'='CHANTRY'
'presents' is the link.
(Other definitions for chantry that I've seen before include "Section of church" , "place for sung mass" , "Chapel for requiem masses" .)