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 categoriesbetween 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 $\Gamma_*(\mathcal{H}om_\mathcal{B}(Y,Z))\cong \hom_\mathcal{A}(\Gamma_*(Y),\Gamma_*(Z))$?