What will happen if we plug a laptop C charger to a mobile? Will it go into fast charge mode ?
There are two scenarios, each with their own result:
1. Charger and device support compatible fast charging standard - this can be something like Qualcomm QC or USB-C PD. In this case, the devices will negotiate the power to be provided. For instance, if I use a 65w charger which supports 18w profile over USB-C PD to charge my Pixel 4a, which also supports USB-C PD, then it will provide 18w power and phone will fast charge
2. Charger and device do not support compatible fast charging standard - suppose charger only supports warp charge and phone only supports USB-C PD. They won't be able to negotiate higher power, but they'll know which is charger and which is capable of delivering power. So charging will happen at 5v 1Amp or 5v 2Amp (5w or 10w) slow charging rates.
In any case, a high powered charger will never damage a low powered device, unless it's a poorly manufactured or faulty device.