Making a GUI element follow the cursor without delay.

Hi all, I want to make a UI element (Several UI elements, actually) follow the mouse cursor without any delay.

If I just do something like

Image UIElement;

void Update(){
    UIElement.transform.position = Input.mousePosition;
}

The UIElement follows the cursor, but if I move the cursor a bit fast it drags behind it a bit. Anyone know a solution?

And no, I cannot just set my element as a custom cursor (it is not actually the cursor, I just want it to always be next to it).

Well, hiding the cursor and using a texture works well enough. Putting this here so I can mark this question as answered.

Maybe you could add a smoothing variable. This could work.