There's something bad going on here, Keith. You told me that \$$\mathbf{N}\$$ has just one object. So, I pointed out that any functor \$$F : \mathbf{Set} \to \mathbf{N}\$$ must send every object of \$$\mathbf{Set}\$$ to that one object, and I wrote:

> So, if you were trying to use such a functor to count the number of elements in sets, it's not going to work.

And yet you seem to be trying to do just that! In your formula you seem to be sending the objects of \$$\mathbf{Set}\$$ to _morphisms_ of \$$\mathbf{N}\$$. That's called a "level slip".

You may want to change your definition of \$$\mathbf{N}\$$ and see if that helps. But I believe you'll have trouble sending both the objects and morphisms of \$$\mathbf{Set}\$$ to something involving natural numbers - even if we restrict attention to finite sets, as you're secretly doing.

I think the solution is to use [decategorification](https://ncatlab.org/nlab/show/decategorification) and note that the decategorification of \$$\mathbf{FinSet}\$$ is \$$\mathbb{N}\$$, the usual set of natural numbers.