Flyer’s angle accepted by two monarchs (10)
I believe the answer is:
kingfisher
'flyer's' is the definition.
(type of bird, something which flies)
'angle accepted by two monarchs' is the wordplay.
'two monarchs' can mean two replacements for 'monarch'.
'angle' becomes 'fish' (synonyms).
'accepted by' means one lot of letters goes inside another.
'monarch' becomes 'king' (king is a kind of monarch).
'monarch' becomes 'ER' (abbreviation for Elizabeth Regina, British Queen).
'king'+'er'='kinger'
'fish' going within 'kinger' is 'KINGFISHER'.
(Other definitions for kingfisher that I've seen before include "Blue bird of the river" , "Colourful river bird" , "Creature" , "Colourful bird seen by streams and rivers" , "Halcyon" .)