In general, the statement is correct. Most of a firm's supply curve is the same as its marginal costs curve. This is because a firm should always produce its product at the quantity where the marginal cost is equal to the marginal revenue.
However, there is one other consideration. The firm must also consider its average variable costs (AVC). If the marginal revenue that can be gained by selling an extra unit of product is below the AVC, the firm should shut down. It will actually lose less money by shutting down than it will by selling the next unit.
So, a firm's supply curve depends on its marginal costs curve, but only above the point where marginal revenue equals average variable costs.