This is a slight reformulation of exercise II.5.9.(c) in Hartshorne's "Algebraic Geometry" which I don't understand.



> 
> Let $K$ be a field and $S=K[X\_0,\ldots,X\_n]$ a graded ring. Set $X=Proj(S)$ and let $M$ be a graded $S$-module. The functors $\Gamma\_\*$ definied by
>  $$
> \Gamma\_\*(\mathcal{F})=\bigoplus\_{n\in\mathbb{Z}} (\mathcal{F}(n))(X)
> $$
>  and $~\widetilde{\phantom{\cdot}}~$ (the "graded associated sheaf functor", see Hartshorne II.5. page 116 for a definition)
>  induce an **equivalence of categories** between the category $\mathcal{A}$ of quasi-finitely generated (i.e. in relation to a finitely generated module) graded $S$-modules modulo a certain equivalence relation $\approx$ and the category $\mathcal{B}$ of coherent $\mathcal{O}\_X$-modules. The equivalence relation is: $M\approx N$ if there is an integer $d$ such that $\oplus\_{k\geq d}M\_k\cong\oplus\_{k\geq d}N\_k$.
> 
> 
> 


I don't know what an "equivalence of categories" is in this context. Formally an "equivalence of categories" means in particular that there are isomorphisms $$\hom\_\mathcal{A}(M,N)\cong \hom\_\mathcal{B}(\widetilde{M},\widetilde{N})$$ and $$\hom\_\mathcal{B}(Y,Z)\cong \hom\_\mathcal{A}(\Gamma\_\*(Y),\Gamma\_\*(Z))$$
of **sets**. This is my problem: How is the sheaf $\mathcal{H}om\_\mathcal{B}(Y,Z)$ considered as a **set**? Perhaps it should be $\Gamma\_\*(\mathcal{H}om\_\mathcal{B}(Y,Z))\cong \hom\_\mathcal{A}(\Gamma\_\*(Y),\Gamma\_\*(Z))$?



Question on an exercise in Hartshorne: Equivalence of categories
The homomorphisms in the category of sheaves are not sheaves themselves. The hom sheaves have the data of things that are only homomorphisms over open subsets. So if $Y,Z$ are coherent $\mathcal{O}\_X$-modules and you are looking for $\mathcal{O}\_X$-module homomorphisms, you don't actually get $\mathcal{H}om(X,Y)$, what you actually get are the global sections only, because these are the only homomorphisms that are defined on the whole space, and so the only actual homomorphisms in the category of sheaves.


