Is this usage of MSELoss correct?

crit = nn.MSELoss(reduction='sum').to(setting.device)
# Do I need to define two criterions 'crit1' and 'crit2'?
# Or just only one criterion 'crit' as below?
extra_loss += crit(A, target1.detach())
extra_loss += crit(B, target2.detach())

Any help is appreciated.

One is enough. The loss here has no (learnable) parameter (in case you define it as a module), it is just a function (F.mse_loss).

By the way, when you define it as a pytorch module, it is always F.mse_loss that is called in forward : pytorch/ at master · pytorch/pytorch · GitHub
One of the purposes of the module is that you can do"device_name") as you did.