Let's try an example 食べます・食べるーー＞ is to eat 食べて <- if you have て　at the end you can add more words.　食べて飲みました I ate AND drank. 食べています・ I'm eating. 食べていて・　I'm eating AND something something.

If you know the progressive form of verb which adds +ている to the verb, on your example, -て is added to the progressive to make it conjugated progressive verb in a sentence. i.e. 求める　＝　求めている(progressive) = 求めていて(conjugated)

the いて is precent progress form. usually ,after 求めていて ,you use continued sentence. sorry,I can't speak English well