# a problem on functor

Let $I$ be a small category and suppose $F:C\rightarrow Set$ and $G:I\rightarrow C$ are functors.

(i) How to construct a morphism $\alpha:F(\varprojlim_{i}G(i))\rightarrow \varprojlim_{i}F(G(i))$ using the projections $\varprojlim G(i)\rightarrow G(i)$.

(ii) How to prove that if $F$ is representable then $\alpha$ is an isomorphism?

(iii) What will be the corresponding statement if $F$ is a functor $F:C^{op}\rightarrow Set$?

Thank you for your help and time.

I will write $\lim$ for $\varprojlim$, and $\lim G$ for the limit of the diagram $G : I \to C$.
For (i), note that by definition, there are maps (I wouldn't call them "projections") $\lim G \to G(i)$ for each $i\in I$, and moreover that these maps are natural in $i$, so that for each arrow $i \to j$, we get a commutative triangle. Applying $F$ gives a system of maps $F(\lim G) \to (F\circ G)(i)$, which is also natural in $i$. The desired morphism $\alpha$ is then unique determined by the universal property of $\lim (F\circ G)$.
Suppose now that $F$ is corepresentable, meaning that it is naturally isomoprhism to $\hom_C(f,-)$ for some object $f\in C$. (A better notation is to say that the functor $\hom_C(f,-)$ is corepresented by $f$, so that the contravariant functor $\hom_C(-,f)$ is the representable functor represented by $f$.) The universal property of the limit implies then that $\hom_C(f,\lim G) =$ the set of systems of morphisms $f \to G(i)$ that are natural in $i$; but such a system is exactly a system of elements in $\hom_C(f,G(i))$, which is to say an element of $\lim \hom_C(f,G)$. The corollary is that corepresentable functors distribute over (i.e. commute with) limits. It is worth thinking about this until it becomes trivial.
Finally, limits in $C^{\mathrm{op}}$ are the same as colimits in $C$. Thus for any contravariant functor $F$, there is a canonical map $F \operatorname{colim} G \to \lim FG$, which is an isomorphism if $F$ is representable.
• A set-valued functor $F$ is corepresentable if there is an initial representable functor $H$ with a morphism $F \to H$ (not vice versa which would lead to representability) and usually in moduli spaces one demands that $F(k) \to H(k)$ is an isomorphism for all fields $k$. – Martin Brandenburg Dec 5 '13 at 0:58