Name of man that's caught nits brought up (5)

I believe the answer is:
cecil
'name of man' is the definition.
'Cecil' can be an answer for 'man' (I've seen this before). I am not sure about the 'name of' bit.
'caught nits brought up' is the wordplay.
'caught' becomes 'c' (cricket abbreviation).
'nits' becomes 'lice'.
'brought up' says the letters should be written backwards (in a down clue, letters go up).
'lice' written backwards gives 'ecil'.
'c'+'ecil'='CECIL'
'that's' is the link.
Can you help me to learn more?
(Other definitions for cecil that I've seen before include "Marquis of Salisbury family name" , "- Day-Lewis, Eng. poet" , "Nobleman" , "Boy" , "___ Day Lewis, Poet Laureate, d. 1972" .)