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

Yes. Because of daylight saving time the gap moves between +1h and +2h.

When is the best time to call Lagos from Cairo?

Between 11:00 and 17:00 in Cairo, which is 09:00 to 15:00 in Lagos.

Cairo is 2h ahead of Lagos. This gap is not fixed: daylight saving time moves it between +1h and +2h during the year.

Cairo

Egypt

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Lagos

Nigeria

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A 9-to-5 day in each city overlaps for 6h. That is 11:00–17:00 in Cairo, which is 09:00–15:00 in Lagos.

City	Time zone	UTC offset
Cairo · Egypt	Africa/Cairo	UTC+3
Lagos · Nigeria	Africa/Lagos	UTC+1

What is the time difference between Cairo and Lagos?: Cairo is 2h ahead of Lagos. This gap is not fixed: daylight saving time moves it between +1h and +2h during the year.
Does the time difference between Cairo and Lagos change during the year?: Yes. Because of daylight saving time the gap moves between +1h and +2h.
When is the best time to call Lagos from Cairo?: Between 11:00 and 17:00 in Cairo, which is 09:00 to 15:00 in Lagos.