Michael wrote:

> I don't see how \\( \text{Ob}: \mathbf{Cat} \to \mathbf{Set} \\) can be a functor if it forgets all the arrows and keeps only the objects. Wouldn't this not be able to preserve composition since it breaks all the objects into separate entities? I think I am missing something here...

I hope Anindya cleared it up for you in [comment #25](https://forum.azimuthproject.org/discussion/comment/19808/#Comment_19808). It seems you were making a "level slip", mixing up the arrows in each object of \\(\mathbf{Cat}\\) (which the functor \\(\text{Ob}\\) simply throws out) with the arrows in \\(\mathbf{Cat}\\) (which the functor \\(\text{Ob}\\) _converts_ from functors to functions).

It's easy to make level slips in category theory, _especially_ when discussing the category of categories, which is practically an _invitation_ to make level slips. So, don't feel bad - just learn to notice when you're slipping: there's a certain sense of confusion involved, which is a sign to slow down and think carefully.

In the paragraph before the last one I was trying to write clearly. If I were trying to torture you further, I might have said something like this:

> It seems you were mixing up the morphisms in each object of the category of categories, which are categories, with the morphisms in the category of categories, which are functors. The functor \\(\text{Ob} \\) from the category of categories to the category of sets maps categories to sets by discarding the morphisms in each category and retaining just their set of objects, and it maps functors to functions by retaining only the functor's action on the set of objects.

Fun!

> I don't see how \\( \text{Ob}: \mathbf{Cat} \to \mathbf{Set} \\) can be a functor if it forgets all the arrows and keeps only the objects. Wouldn't this not be able to preserve composition since it breaks all the objects into separate entities? I think I am missing something here...

I hope Anindya cleared it up for you in [comment #25](https://forum.azimuthproject.org/discussion/comment/19808/#Comment_19808). It seems you were making a "level slip", mixing up the arrows in each object of \\(\mathbf{Cat}\\) (which the functor \\(\text{Ob}\\) simply throws out) with the arrows in \\(\mathbf{Cat}\\) (which the functor \\(\text{Ob}\\) _converts_ from functors to functions).

It's easy to make level slips in category theory, _especially_ when discussing the category of categories, which is practically an _invitation_ to make level slips. So, don't feel bad - just learn to notice when you're slipping: there's a certain sense of confusion involved, which is a sign to slow down and think carefully.

In the paragraph before the last one I was trying to write clearly. If I were trying to torture you further, I might have said something like this:

> It seems you were mixing up the morphisms in each object of the category of categories, which are categories, with the morphisms in the category of categories, which are functors. The functor \\(\text{Ob} \\) from the category of categories to the category of sets maps categories to sets by discarding the morphisms in each category and retaining just their set of objects, and it maps functors to functions by retaining only the functor's action on the set of objects.

Fun!