It is the auxiliary verb 'does' (with an 's' at the end) which agrees with the pronoun 'he'. The main verb (have) is it the infinitive form: 'have'. When you have the auxiliary 'do'/'does'/'did', or their negative forms, in your sentence, the main verb doesn't have an 's'. For example: Affirmative --> He goes. Negative ---> He doesn't go. Question ---> Does he go? Affirmative --> He has. Negative ---> He doesn't have. Question ---> Does he have?