Sensitive crossword setter's embracing one very loudly (5)

'sensitive' is the definition.
(thesaurus)

'crossword setter's embracing one very loudly' is the wordplay.
'crossword setter's' becomes 'my' (belonging to the crossword setter).
'embracing' indicates putting letters inside.
'one' becomes 'i' (Roman numeral).
'very loudly' becomes 'ff' (short for fortissimo, 'very loud' in music).
'i'+'ff'='iff'
'my' enclosing 'iff' is 'MIFFY'.