The authors begin by defining the concept of a left natural number object (LNNO) in a monoidal category. If it exists it is unique up to isomorphism. A right natural number object (RNNO) is defined in a similar way. If both exist then they are isomorphic. The authors also gives an example of an LNNO which is not RNNO. It is shown that an LNNO can be made into a monoid. In the second section the authors construct morphisms \(N^ k\otimes A\to A\otimes N^ k\) where N is an LNNO and shows that the full subcategory determined by the objects isomorphic to \(N^ k\) is a symmetric monoidal category. This simplifies the study since there is no distinction between LNNO and RNNO if we have symmetry. In that case we let NNO denote LNNO or RNNO. In the third section the authors show that if N is an NNO then N has a comonoid structure which is cocommutative. It is a also noted that the category of cocommutative comonoids is cartesian. This makes it possible to reduce the study of NNO's to the case where the category is cartesian where the subject is well developed. (This is dealt with in a paper which has not yet appeared.) The final section deals with free monoidal categories with LNNO.