It's right and appropriate to be thoroughly engrossed (4)
I believe the answer is:
rapt
'thoroughly engrossed' is the definition.
'rapt' can be an answer for 'engrossed' (synonyms). I am unsure of the 'thoroughly' bit.
'it's right and appropriate' is the wordplay.
'it's right' becomes 'r' (abbreviation).
'and' means one lot of letters go next to another.
'appropriate' becomes 'apt' (synonyms).
'r'+'apt'='RAPT'
'to be' is the link.
Can you help me to learn more?
(Other definitions for rapt that I've seen before include "Very attentive" , "Part (anag.)" , "Spellbound, enthralled" , "Taken up entirely" , "Fully intent, fascinated" .)