Category of accessible functors and its closedness

Question

Is the category of $\sf{Set}$ accessible endofunctors right closed w.r.t. composition (as a monoidal structure)? Any hint on how to prove this?

I think that this is true if one works with finitary endofunctors (formula 4.6 of Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads, by Kelly and Power).

Answer 1

Let $\kappa$ a regular cardinal and let $\mathcal{M}_\kappa$ be the category of $\kappa$-accessible functors $\mathbf{Set} \to \mathbf{Set}$.

1. $\mathcal{M}_\kappa$ is a finitely accessible category: indeed, $\mathcal{M}_\kappa$ is equivalent to $[\mathbf{Set}_{< \kappa}, \mathbf{Set}]$
2. The inclusion $\mathcal{M}_\kappa \hookrightarrow [\mathbf{Set}, \mathbf{Set}]$ preserves colimits for arbitrary small diagrams.
3. For any functor $F : \mathbf{Set} \to \mathbf{Set}$, ${-} \circ F : [\mathbf{Set}, \mathbf{Set}] \to [\mathbf{Set}, \mathbf{Set}]$ preserves colimits for arbitrary small diagrams.
4. Hence, ${-} \circ F : \mathcal{M}_\kappa \to \mathcal{M}_\kappa$ preserves colimits for arbitrary small diagrams, and therefore has a right adjoint by the accessible adjoint functor theorem.

The problem is this: if $\lambda$ is a regular cardinal and $\kappa < \lambda$, then $\mathcal{M}_\kappa \subseteq \mathcal{M}_\lambda$, but I see no reason for the inclusion to preserve the right adjoint of ${-} \circ F$. (It's hard enough to understand what the right adjoint is!) Assuming that is true, and that is a big assumption, then it would be true that the category $\mathcal{M} = \bigcup_\kappa \mathcal{M}_\kappa$ of all accessible functors $\mathbf{Set} \to \mathbf{Set}$ is monoidal closed.