Phrase(s): babysit for someone [and] babysit (with) someone
1. to attend and care for a child for a period of time. • I’m looking for someone to babysit for my cousin. • Will you babysit with my cousin? 2. to attend and care for a child for someone for a short period of time. • Would you mind babysitting Roger for me for a few minutes? • Sure, I will babysit for you.
McGrawhill's American Idioms And Phrasal Verbs