The Mexican government will eventually extradict EL Chapo to the US. For one thing, it will improve relations between our two countries. Also Mexico will have to worry less about violence surrounding the trial. Finally El Chapo is more likely to be convicted in the US, will the spectre of corruption is less likely.
I do not believe Joaquin "El Chapo" Guzman will be extradited to the United States. The Mexican government caught him and they have plenty of charges against him to keep him locked up for the rest of his life. There is no reason for the United States to request his presence or take on the expense of housing him.
I believe Joaquin Guzman will stand trial in Mexico, but I seriously don't see him being extradited to the United States. Even though he's in custody in Mexico, I would guess it's still business as usual for his drug cartel. I would imagine they have the power and the money to keep the courts and legal process in limbo for a number of years.
There is far too much corruption in Mexico for El Chapo to ever be taken to the United States for judgment and trial. For all we know, he probably planned to get captured and has a plan to escape from now on. He is a mastermind, with the Mexican nationals behind him.
El Chapo was captured in Mexico and because of that, he will remain in Mexico and not be taken to America. The Mexican officials want to try him in their country for crimes he committed against the Mexican people. America would also like to try him for the many crimes he did here.