Stitching in this case is something that is done to the background sprite. It basically means that the left edge of the sprite matches up with the right edge, so the background still looks good when the player walks to the edge of it , and you see two instances of the background, one at the left edge and one with the right edge lined up so you can't see the seam.
This seems like an interesting challenge. If you do end up moving the background instead of the player, you have to move all the obstacles and enemies etc also. What I would do instead is designate a margin the width of the screen around the edge of the layout. For everything within that margin, you duplicate it so that everything that happens on the right happens on the left and visa versa. If the duplication is good enough, the player won't notice when they are teleported from one edge to the other. Then you modify all your pathfinding and enemy behaviors so they respect the boundary as well, and make sure enemies also get teleported when they hit the edge. I bet that using a Canvas can help you to visually duplicate parts of the layout, which makes it easier to hide that seam.