Let $X$ be a smooth variety and $M$ a fine moduli space of certain kind of sheaves on $X$. Let $\mathcal{E}$ be the universal family on $X\times M$. Suppose there is a derived functor $F$ from $D^b(X)$ to some derived category, say $D^b(Y)$, or $D^b(A-mod)$ for some finite algebra $A$, so that $F(\mathcal{E}_t)$ is a semistable object of some kind (say semistable sheaves on $Y$ or semistable rep of $A$). This provides a map for $M \to N$ where $N$ is the moduli of semistable objects on the target category. 

My question: is this map always algebraic or is there any criterion for this map to be algebraic?

Thank you very much for your help.