The cofree coalgebra using adjoint functor theorems

Let $k$ be a commutative ring. There is a forgetful functor $$U : \mathsf{Coalg}_k \to \mathsf{Mod}_k$$ from $k$-coalgebras to $k$-modules. This has a right adjoint, called the cofree coalgebra on a $k$-module (for instance, see section 1.6 in the book "Hopf algebras" by Dascalescu, Nastasescu, Raianu).

Question 1. In Michael Barr's paper "Coalgebras over a commutative ring", it is shown that $U$ has a right adjoint using the special adjoint functor theorem. In order to apply this, $\mathsf{Coalg}_k$ should have a small genenerating set. Why does Barr say nothing about this? And how to prove that $\mathsf{Coalg}_k$ has a small generating set? If $k$ is a field (more generally, when $k$ is absolutely flat), this follows from the fundamental theorem on coalgebras.

Question 2. Can we prove the existence of the cofree coalgebra using the general adjoint functor theorem? For this, we would have to check that for every $k$-module $M$ the category $U \downarrow M$ has a weakly terminal set of objects. But I could not find it so far.

Question 3. (Edit: moved to math.SE/1631890)

Question 4. More generally, let $(\mathcal{C},\otimes)$ be a cocomplete monoidal category (i.e. $\mathcal{C}$ is cocomplete and $\otimes$ is cocontinuous in each variable). Under what conditions does the forgetful functor $\mathsf{Coalg}(\mathcal{C},\otimes) \to \mathcal{C}$ have a right adjoint?

• I think it would be a good idea to split up these four questions into at least 2-3 questions... Especially now that (as far as I understand, sorry if I understood incorrectly) there is an answer that only addresses two of the four questions. If necessary you can link the questions to one another. Question 3 in particular seems quite independent from the others. – Najib Idrissi Jan 29 '16 at 11:04
• I will ask Q3 separately. – Martin Brandenburg Jan 29 '16 at 11:39
• In Q2, shouldn't you be looking for a weakly terminal set of objects? – Zhen Lin Jan 29 '16 at 11:56
• I have corrected it. – Martin Brandenburg Jan 29 '16 at 14:07

If $\mathcal{C}$ is an accessible category and $\otimes$ is an accessible functor, then $\mathbf{Coalg} (\mathcal{C})$ is accessible. This is a special case of the theorem on 2-limits of accessible categories; after all, $\mathbf{Coalg} (\mathcal{C})$ can be constructed using comma categories, products, iso-inserters, and equifiers. (This is the same argument used to show that the category of (co)algebras for an accessible (co)monad is accessible.) This yields the desired generating set.
In particular, if $\mathcal{C}$ is a locally presentable monoidal closed category, then $\mathbf{Coalg} (\mathcal{C})$ is also locally presentable and the forgetful functor $\mathbf{Coalg} (\mathcal{C}) \to \mathcal{C}$ has a right adjoint.
• I suppose it's possible in principle. If you know that $\mathbf{Coalg} (\mathcal{C})$ is $\kappa$-accessible then the $\kappa$-presentable objects form a dense subcategory. The problem is that, in the standard textbooks, no estimate of $\kappa$ is given – you have to chase through the proofs carefully. However, you might have better luck looking at the 1977 preprint of Ulmer (check your email). – Zhen Lin Jan 29 '16 at 10:05