That may not be the first thing that comes to mind when you consider leadership.
The word that is translated “build up” is οικοδομη (oikodomē). You can get a sense of how the term is used by looking at how it is translated in the New Testament. Here are all 18 of the instances of the use of οικοδομη.
If Saul had had the attitude that God gave him authority to build up Israel, perhaps he would have worked at building up David rather than trying to kill him out of jealousy.
Jealousy, comparison, and competition have damaged the ministries of many leaders. We must both be aware of this and also watch for it in our leaders. A cursory reading of 2 Corinthians reveals that Paul used his authority in a selfless way that demonstrated his exhortation to the Philippian church in Philippians 2.
It is not natural for us to respond in this manner. It requires the grace of God active in our lives. It requires abiding in Him, John 15:5.