# Presheaf and copresheaf categories on finite sets

Briefly: do they agree?

In more detail: denote by $\mathbf{Finset}$ the category of finite sets, and by $\mathbf{Set}$ the category of sets. I want to know what the functor categories $[\mathbf{Finset},\mathbf{Set}]$ and $[\mathbf{Finset}^{\text{op}},\mathbf{Set}]$ look like. Are they both equivalent to $\mathbf{Set}$?

How should I think about these so that I at least conjecture the answer (and then try to construct a functor that is an equivalence)? I am really struggling with thinking about (explicit) functor categories intuitively.

• $\mathbf{Set}$, $[\mathbf{FinSet}, \mathbf{Set}]$ and $[\mathbf{FinSet}^\mathrm{op}, \mathbf{Set}]$ are all distinct. – Zhen Lin Apr 22 '16 at 13:42

A set $X$ determines very special functors from finite sets to sets, contravariantly by sending $S$ to the set of functions $f:S\to X$ and, dually, covariantly. These are restrictions of representable functors on the full category of sets, so for instance the contravariantly one send colimits to limits. There's no reason at all why a general functor should do that-how about the constant functor $S\mapsto \{a,b\}$?