Does the time difference between Lagos and Sydney change during the year?

Yes. Because of daylight saving time the gap moves between −10h and −9h.

When is the best time to call Sydney from Lagos?

There is no overlap between standard working days, so one side has to call early or late.

Lagos is 9h behind Sydney. This gap is not fixed: daylight saving time moves it between −10h and −9h during the year.

Lagos

Nigeria

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Sydney

Australia

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A 9-to-5 working day in Lagos does not overlap one in Sydney at all — somebody has to take the call outside office hours.

City	Time zone	UTC offset
Lagos · Nigeria	Africa/Lagos	UTC+1
Sydney · Australia	Australia/Sydney	UTC+10

What is the time difference between Lagos and Sydney?: Lagos is 9h behind Sydney. This gap is not fixed: daylight saving time moves it between −10h and −9h during the year.
Does the time difference between Lagos and Sydney change during the year?: Yes. Because of daylight saving time the gap moves between −10h and −9h.
When is the best time to call Sydney from Lagos?: There is no overlap between standard working days, so one side has to call early or late.