Ill-disciplined pet-lovers — missed taking dog for morning walk? (9)

I believe the answer is:

overslept

I'm a little stuck... Click here to teach me more about this clue!

'dog for morning walk?' is the definition.
I can't judge whether this definition defines the answer.

'ill-disciplined pet-lovers missed' is the wordplay.
'pet-lovers' anagrams to 'OVERSLEPT'.
Though, I'm not sure how the anagram is indicated.

'taking' acts as a link.

Can you help me to learn more?

(Other definitions for overslept that I've seen before include "Was late rising" , "Failed to rise on time" .)

I've seen this clue in The Times.

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