Cold inside suet, that's true (4)

I believe the answer is:
fact
'true' is the definition.
Although both the answer and definition are singular nouns, I cannot understand how they can define each other.
'cold inside suet' is the wordplay.
'cold' becomes 'C' (eg on taps).
'inside' is an insertion indicator.
'suet' becomes 'fat' (I've seen this in another clue).
'c' inserted inside 'fat' is 'FACT'.
'that's' acts as a link.
Can you help me to learn more?
(Other definitions for fact that I've seen before include "matter of record" , "Statement of verifiable information" , "Actuality" , "Undisputed truth" , "Something indisputable" .)
