Fighting a southpaw
I fucking hate boxing southpaws, just thinking about it makes me angry.
But here are some things that might help:
-The jab is less effective against them, so unless you have a large reach advantage, don't rely on it too much, as it gives them countering opportunities. Their forward hand is right in front of yours, that's why they can intercept a jab easily.
-Straight right hand and left hook leads can work really well, for the same reasons jabs do not work well: the right hand comes from the other side, so gets past his guard easier, and your left can go around his right.
-Adept your guard a bit. Normally, you're using your right hand on the side of your face to block left hooks, but now you need it a bit more in front, to block that damn straight left hand. Also, keep your own left hand a bit less extended to block any right hooks.
Fighting a Mayweather-style
I'm sure the style has been done and named before, but for lack of a better name, i'll refer to it as the Mayweather defensive style:
Keeping the right hand up as well as the left shoulder high so that the face is very well covered as well as the left side of the body. Not many people adopt this style because it requires great skill, calmness under fire and reflexes, but as Floyd showed, it can be very effective.
What works great against it? Combinations. Single shots are easily picked off, but combinations or much harder to defend against.
The right hand to the body is also a great punch because that side usually is a bit exposed, especially while leaning as the picture shows. Best would be combining these two, i.e. a combination that features one or more right hands to the body. Make sure not to over-extend or leaning in because they're always ready to fire a shot.
General advice on lateral movement.
Best is to vary your direction, but you don't want to keep walking into his money punch. So, if you know which hand he throws best with, you can anticipate this in your movement with a really easy rule of thumb: move to the same side from your perspective as his as best side from his perspective. Basically, if he has a great left hook, then you can best move to your left, if he's better with the right hand then you better move to your right, etc.