Why is the supply curve of an individual firm operating under perfect competition given by that firm's marginal cost curve?

In a competitive market, a firm's production decisions is dictated by its desire to maximize profit. Since firms in a competitive market take price as given, profit is maximized when marginal cost is equal to price. Therefore, if we want to know at any given price, how many units a firm will be producing, we can project the price onto the marginal cost curve, and back out the quantity such that marginal cost is equal to price. Hence, the marginal cost curve also indicates the quantity a firm is willing to supply at any given price, i.e., it is the firm's supply curve.