# How many endomorphisms $(\mathbb{N},\times)$ has? Like object in Mon category

I was told that there is only one morphism in $$Mon$$ category for this object $$(\mathbb{N},\times)$$. But why?

I think that we can write every natural number as the product of a certain set of prime numbers. So we can say that the set of primes forms the basis of this monoid $$(\mathbb{N},\times)$$, that is, linear combinations with positive integer coefficients (powers) make up the entire set of supports for the structure of this monoid: for any $$n\in\mathbb{N}$$, $$n = p_1^{g_1}\times\cdots\times p_i^{g_i}$$, where $$\forall p_j \in \{p_1,p_2,\cdots\}$$ -- basis, and $$\forall g_j\in \mathbb{Z_{>0}}$$. And we can make some transpositions on the elements of the basis $$\{p_1,p_2,\cdots\}$$ thereby defining the morphism (where always $$1$$ going to $$1$$ by this morphism). So we have infinite number of morphisms, I am right?

• Are you sure they said $(\mathbb{N},\times)$ and not $(\mathbb{N},+)$? – Eric Wofsey Dec 16 '18 at 2:23
• @EricWofsey ye, multiplication – Just do it Dec 16 '18 at 2:32
• Yes an infinite number of morphisms – BananaCats Category Theory App Dec 16 '18 at 3:43