Interpretation of the loss function for word2vec
Not sure what the video said, but $T$ should not be the vocabulary size, but the training corpus size (number of all words). 

For example, if your training corpus is 

deep learning is popular . i love deep learning . i want to learn more about it.


Then when you sume up over $T$, you will sum up all the word pairs in the corpus including duplicates. The word pair (deep learning) is indeed calculated twice. 

For details please refer to the original skip-gram paper, and notice the definitoin (1) on page 2.