I am reading the book “Fourier-Mukai transforms in algebraic geometry" by Daniel Huybrechts. In the proof of Lemma 4.5 in page 92, he uses
Lemma: If $M$ is a finite module over a local ring $(A,m)$ such that $\operatorname{supp}(M)=\{m\}$, then there exists a surjection $M\twoheadrightarrow A/m$ and an injection $A/m\hookrightarrow M$.
and he concludes that if $\mathcal{F}$ and $\mathcal{G}$ are two coherent sheaves over a smooth projective variety $X$ such that $\operatorname{Supp}(\mathcal{F})=\operatorname{Supp}(\mathcal{G})=\{x\}$, then there exists a homomorphism $\mathcal{F}\to \mathcal{G}$. My question is that why is this true?