Moduli space of pairs over projective stacks

Let $\text{clX}$ a projective stack over an algebraically closed field $k$ of characteristic 0. Let $\text{clE}$ be a generating sheaf over $\text{clX}$ and ${\text{clO}}_X(1)$ a polarization of its coarse moduli space $X$. We define a notion of pair which is the datum of a non vanishing morphism $\mathrm{\Gamma }\otimes \text{clE}\to \text{clF}$ where $\mathrm{\Gamma }$ is a finite dimensional $k$ vector space and $\text{clF}$ is a coherent sheaf over $\text{clX}$. We construct the stack and the moduli space of semistable pairs. The notion of semistability depends on a polynomial parameter and it is dictated by the GIT construction of the moduli space.