db={"Collaborators":[{email:"me@me.com"}],"Initiatives":[{title:"A",team:"1",Collaborators:[{rating:"4",role:"leader",email:"me@me.com"},{rating:"4",role:"participant",email:"not_me@me.com"}]},{title:"B",team:"2",Collaborators:[{rating:"4",role:"leader",email:"not_me@me.com"},{rating:"4",role:"participant",email:"not_me@me.com"}]},{title:"C",team:"3",Collaborators:[{rating:"4",role:"leader",email:"not_me@me.com"},{rating:"3",role:"participant",email:"me@me.com"}]}]}

db.Collaborators.aggregate([{$lookup:{from:"Initiatives",let:{email:"$email"},as:"Initiatives",pipeline:[{$match:{$expr:{$in:["$$email","$Collaborators.email"]}}},{$unwind:"$Collaborators"},{$replaceWith:{$mergeObjects:["$Collaborators",{title:"$title"}]}}]}},{$set:{Initiatives:{$filter:{input:"$Initiatives",cond:{$eq:["$$this.email","$email"]}}}}},{$unset:"Initiatives.email"}])