The way it is at the moment from the photo, water from the sink could get into the washing machine drainage plus (and a big one) you'll have permanent standing water up to the level where the horizontal hose hits the level of the sink outlet....smelly and loads of backwash when you use the machines. Better way would be to lift the level of the machines drainage to the height of the sink trap inlet and to use a double spigot top inlet (like you've shown) onto the single stand pipe plus a trap connection where it joins the horizontal pipe. If that makes sense
Better still would be to have a separate outlet for the sink and another for the d/w and w/m drainage with traps for everything.