The issue here is, only the root user has the permission to write in the given folder. So, as ec2-user you cannot do that.
1. The simplest solution is to copy the same war file to the /home/ec2-user folder and then ssh into the instance and copy the file from your home directory to /var/lib/tomcat6/webapps/
cp -i WARfilename.war /var/lib/tomcat6/webapps
2. The other solution but not suggested is to SSH into the instance, go to the tomcat6 directory and change the permission of the webapps directory:
sudo chmod 777 webapps
Once this is done, and the permissions are changed, you can copy the file directly from Filezilla as the ec2-user.
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