Finitely presentable objects in functor categories

Question

Given a locally finitely presentable category $\mathcal{C}$ it is well-known that every functor category $[\mathcal{A},\mathcal{C}]$ (where $\mathcal{A}$ is a small category) is also locally finitely presentable. Is there a concrete description of the finitely presentable objects in such categories?

If this is not possible in general, what about special cases like $\mathcal{A}=\mathcal{Set}_{fin}^{op}$?

Answer 1

When $\mathcal{A}$ is a finite category, the finitely presentable objects in $[\mathcal{A}, \mathcal{C}]$ are precisely the diagrams that are componentwise finitely presentable: see e.g. Proposition 2.23 here. The general case is harder to describe. There are two steps:

1. First, determine the finitely presentable objects in $[\operatorname{ob} \mathcal{A}, \mathcal{C}]$; these will contain e.g. "finitely supported" families of finitely presentable objects.
2. Then, determine the left adjoint of the restriction functor $[\mathcal{A}, \mathcal{C}] \to [\operatorname{ob} \mathcal{A}, \mathcal{C}]$; by general nonsense, the restriction functor is finitely accessible and monadic, and the finitely presentable objects of $[\mathcal{A}, \mathcal{C}]$ will be the objects in the smallest full subcategory that is closed under finite colimits and that contains the free "algebras" generated by the finitely presentable objects in $[\operatorname{ob} \mathcal{A}, \mathcal{C}]$.