I'm going to the library

[Brightsun]

The following sentence in my English text book describes the picture in which a boy is on the way to the library.

He is going to the library to study for an English exam.

I have thought the sentence is used to mean that he will go to the library

If I am right. I think the sentence should be corrected like this,

He is on the way to the library to study for an English exam.

Am I right? Thank you so much!

[boothling]

Both sentences are correct.