Is the marginal cost the same for every firm in a perfectly competitive market? Spin-off from: market equilibrium quantity $\ne$ firm profit maximising quantity?

Consider a perfectly competitive market with equilibrium price $P_{eq}$ and quantity $Q_{eq}$ and firm with profit maximising quantity $Q_f$ as illustrated below:







Is the MC for each firm in the market the same?

Can we deduce such from any of the assumptions listed in the PC Wiki page?
No, the marginal cost curves are not necessarily the same for each firm in the market. However the values of marginal costs are.

To disprove the general claim that "The marginal cost curve of each firm in a competitive market is the same" we simply need to find one counter-example, such as the one given below:

Suppose there are two firms in the market and the market inverse demand function is given by $P=7 - q^m = 7 - (q_1 + q_2)$

Here $P$ represents the price, $q^m$ the market supply and $q_1, q_2$ represent the respective supplies of the firms 1 and 2.

Suppose the marginal cost curves for each firm are given by:

$MC_1 = 2q_1$,

$MC_2 = 4q_2$

Note that these are not the same! Since we are looking for a competitive equilibrium, the firms behave competitively and hence we must assume that they do not realize they have market power. Hence they each set:

$P = MC_i, \; i\in\{1,2\}$

From this, solving for quantity we have the respective Supply functions of each firm:

$S_1 = q_1 = 0.5P$

$S_2 = q_2 = 0.25P$

Inserting these two supply functions into the inverse demand function we have:

$P=4$, $q_1=2$, $q_2=1$.

Is this an equilibrium with perfect competition? Well we found it by combining Supply and Demand assuming perfectly competitive firms, so it must be. To verify:

$Market \; Demand = 7 - p = 3$

$Market \; Suppy=q_1 + q_2 = 3$

So the market is indeed in equilibrium. Are both firms setting $P=MC$ as they should in perfect competition?

$MC_1 = 2*q_1 = 4 = P$

$MC_2 = 4*q_2 = 4 = P$

So indeed we have a market equilibrium with perfect competition and different marginal cost curves. QED

Note however that the values of the marginal cost curves at the equilibrium quantities are the same for each firm (and must be the same for each firm in equilibrium).