No one asked you to bully, just to support your claims. No universal truths does not mean no need to support your values. What do you think ethical codes are, trading cards?
"Hey look at my morals!"
"Wow those are cool. Do you like mine?"
This is the basis for all of your interpersonal behavoir in your life, and how you affect those around you. If you are unwilling to consider that your morals could be ill-founded, then you are displaying willful ignorance, which makes you a poor souce of any moral insight or input.