The schematic of OPAMP was shown in the attached figure. Besides the main two input transistors, there are two extra cross-coupled transistors with smaller sizes. Does anyone know the function of them? It seems they will reduce Gm of OPAMP.

Thanks for your advice!

To expand on Carl's statement, they keep the cascode transistors biased up even when the opamp is slewing.  

Normally, during slewing, all of the current would be to one side, debiasing the cascode devices.  The cross coupled device steers some current back to the cascode device.

I just look around what in my favorites and found that I can answer the question. The extra cross-coupled pair is for neutralization reducing miller effect. More details of neutralization can be found in the textbook by Paul, Gray, and Meyer.

This is a telescopic amp with gain-boosted n-cascodes. The purpose of level-shift circuit is to allow more headroom for the main amp.. Without it, additional Vt will appear in the headroom requirement of the input diff pair. Instead of 2Vdsat, we would need 2Vdsat+Vt to keep input diff pair and tail current source of the main amp in saturation, hence additional Vt reduction in the available output swing.

The progressive sizing of the bottom two triode transistors in "nVreff generator" on the far left is to account for backgate bias for the stacked transistor with size (1/3) to make it slightly stronger than the one below it due to its larger Vt. That's why they got the same Vds = 400mV.

And I agree with the functions of the small cross-coupled pair as explained by the previous posts above.

I do not quite understand why the level shifter can increase output swing. The level shifter can set the voltage @ the drain of input pair to be 2Veff and a biasing CKT can do that too. I was wondering why a term Vt comes up without a level shifter.