Question over wording in textbook

So I tried asking this question in Physics Stackexchange and got every response except the one that answerd my question :(

Anyway, my question regards this statement i found in my physics textbook: "no work is required to move a charge from one point to another on an equipotential line"

Do they mean work by the electric field or work by anything? Because clearly the object cant just magically move with nothing.

For example, if i place a positive charge in an electric field it clearly moves radially outwards. Now if we pause time (for a bit after it moves) and consider how to move that charge from one point in the equipotential line (that it is currently occupying) to another in the same line, we see that it would require moving the charge perpendiculat to its current position. Now how can this be done with no work? Clearly we need to change its momentum/path to be perpendicular and that would require some type of force and a distance right (and as a result some work is being done to the charge)? This leads me to infer that they meant the electric field when they said this, is that right?