I am trying to understand what exactly general covariance states. As I understand general covariance appeared as generalization of relativity principle so I will try to state relativity principle in a manner that I consider more convenient for my purpose. So let's say we have inertial coordinate system K and in that coordinate system we have coordinate dependent formulation of physical law A. Now in a certain way we transform inertial coordinate system K into inertial coordinate system K' and in that new coordinate system we have coordinate dependent formulation of physical law A' that takes the same mathematical form as law A in K. Relativity principle states that if law A in K holds then law A' in K' holds as well. And we can experimentally test this statement. We take well tested physical law A in K then find K', formulate A' and then translate K' back into K along with physical law A' so that we get physically identical coordinate dependent law B as law A' but in different mathematical form as law A. Within coordinate system K we test law B and if it holds we say that relativity principle holds. So we can use relativity principle to formulate new coordinate dependent physical law B in K if we have coordinate dependent physical law A in K. This might not be very popular formulation of relativity principle but nonetheless just as valid. And the point of this formulation is that relativity principle leads to a new physical laws within single coordinate system. Now the question about generalization of relativity principle to general covariance. In what sense relativity principle is generalized to arrive at general covariance? I would imagine that general covariance applies to coordinate dependent laws in any coordinate system (with primary interest in accelerated coordinate systems) if we have such laws (that are most conveniently formulated in accelerated coordinate system). And then transforming this coordinate system in a certain way we arrive at new laws (with the same mathematical form as primary law) that we think will hold in this new coordinate system. And of course we can translate it back into original coordinate system and get new laws in the same coordinate system (but expressed in different mathematical form as primary law). Does this seem correct?