Functors (between oo-categories) that aren't strictly unital

Let C and D be quasi-categories (a.k.a. infinity-categories). Suppose also that one has an assignment sending commutative diagrams of C to commutative diagrams of D which respects face maps, but not necessarily degeneracy maps. (This is akin to having an assignment which respects all compositions, but may not send identity morphisms to identity morphisms.) When does this assignment give rise to an actual functor? We show that if the original assignment can be shown to respect identity morphisms up to homotopy, then there exists an honest functor of infinity-categories which respects the original assignments up to homotopy. Moreover, we prove that such honest functors can be chosen naturally with respect to the original assignments.