Cat rising in the morning, out of bed (4)
I believe the answer is:
puma
'cat' is the definition.
(puma is a kind of cat)
'rising in the morning out of bed' is the wordplay.
'rising' says the letters should be written in reverse (going up in a down clue).
'in the morning' becomes 'am' (AM on a clock).
'out of bed' becomes 'up' (synonyms).
'am'+'up'='amup'
'amup' back-to-front is 'PUMA'.
(Other definitions for puma that I've seen before include "South American cat" , "South American carnivore" , "Cougar" , "American wildcat" , "Mountain lion" .)