Maybe let me explain Sasha's answer in more detail.

The moduli space $M:=\overline{M_v(X)}=\overline{M^s_v(X)}$ is the coarse moduli space corepresenting the functor $$\mathfrak{M}\colon ({\bf Sch }/\mathbb{C})^{\bf op }\rightarrow{\bf Sets}$$ such that $$\mathfrak{M}(T)=\{\mathcal{F}\in{\bf Coh}(X\times T)\,|\,\mathcal{F}_t\text{ is } H\text{-stable on }X\text{ and }ch(\mathcal{F}_t)=v\text{ for each }t\in T\}/\cong$$ In particular, there exists a natural transformation $\alpha\colon\mathfrak{M}\rightarrow\hom(-,M)$. Hence the morphism $i\colon X\rightarrow M$ is defined as the image of the family $\mathcal{K}\in \mathfrak{M}(X)$ under $\alpha_X$. Moreover, one checks that $$i\colon X\rightarrow M,\quad x\mapsto [\mathcal{K}_x]$$ according to the naturality of $\alpha$.

Maybe let me explain Sasha's answer in more detail.

The moduli space $M:=\overline{M_v(X)}=\overline{M^s_v(X)}$ is the coarse moduli space corepresenting the functor $\mathfrak{M}\colon ({\bf Sch }/\mathbb{C})^{\bf op }\rightarrow{\bf Sets}$ given by $$\mathfrak{M}(T)=\{\mathcal{F}\in{\bf Coh}(X\times T)\,|\,\mathcal{F}_t\text{ is } H\text{-stable on }X\text{ and }ch(\mathcal{F}_t)=v\text{ for each }t\in T\}/\cong$$ In particular, there exists a natural transformation $\alpha\colon\mathfrak{M}\rightarrow\hom(-,M)$. Hence the morphism $i\colon X\rightarrow M$ is defined as the image of the family $\mathcal{K}\in \mathfrak{M}(X)$ under $\alpha_X$. Moreover, one checks that $$i\colon X\rightarrow M,\quad x\mapsto [\mathcal{K}_x]$$ according to the naturality of $\alpha$.