Let $X$ be a curve. Consider the category which has as objects triples $(S,F,\alpha)$ where $S$ is a finite set of closed points in $X$, $F$ is a vector bundle on $X$, and $\alpha$ is a trivialisation of $F$ outside $S$. As morphisms we take $$Hom((S,F,\alpha),(T,G,\beta)):=\{S\subset T, \phi:F|_{X\setminus (T\setminus S)}\xrightarrow{\sim} G|_{X\setminus (T\setminus S)}\mbox{commuting with $\alpha$ and $\beta$ on $X\setminus T$}\}.$$
Suppose that $S$ and $S'$ are disjoint sets of closed points of $X$, and to have triples $(S,F,\alpha)$, $(S',F',\alpha')$. Then we can use the trivialisations to build a gluing of $F$ and $F'$, since the two sets are disjoint.
Is this a coproduct in the category defined above? In fact, having a map from the two objects to another object $(T,G,\beta)$ is to have isomorphism between $G$ and $F$ outside $S'$, and between $G$ and $F'$ outside $S$. I would like to conclude invoking the uniqueness of the gluing of sheaves, but... does this suffice? I fear that there is some problem with the isomorphism, which could not be unique...
Thank you for any help.