second statement of GAGA GAGA theorem 2 states that:
If $\mathscr{F}$ and $\mathscr{G}$ are two coherent algebraic sheaves on X, every analytic homomoprhsim of $\mathscr{F}^h$ into $\mathscr{G}^h$ comes from a unique algebraic homomorphism of $\mathscr{F}$ into $\mathscr{G}$.
What are the homomorphisms between $\mathscr{F}$ and $\mathscr{G}$? Can we just take global sections of the sheaf $Hom_\mathcal{O}(\mathscr{F},\mathscr{G})$ 
see the main statement section of this article:
https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry

If $X$ is a scheme over $\mathbb{C}$ then an algebraic homomorphism between $\mathcal{F}$ and $\mathcal{G}$ is a $\mathcal{O}_{X}$-linear homomorphism of the sheaves $\mathcal{F}$ and $\mathcal{G}$ which is given by a compatible collection of homomorphisms between $\mathcal{G}(U)$ and $\mathcal{F}(U)$ (or if you wish, a natural transformation of the functors $\mathcal{F}$ and $\mathcal{G}$).
Now there is a way of passing to the analytification of the sheaves (passing from $X$ to the complex valued points, holomorphics functions etc.). The GAGA theorem asks whether every homomorphism of ther analytification comes from an algebraic one, and in the case of sufficiently nice ones (coherent) answers this question by "yes".
n.B.: GAGA is standard terminology in algebraic geometry.
