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. 
 A: 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.
