Let $f: X \to S$ be a projective morphisms between finite-type projective schemes, and $\mathcal{O}_{X}(1)$ an $f$-ample line bundle. Given an $S$-flat coherent $\mathcal{O}_{X}$-module $\mathcal{H}$ and a polynomial $P$, we can form the relative Quot-scheme $\pi : Q = \text{Quot}_{X/S}(\mathcal{H},P) \to S$ parameterizing quotients $\mathcal{H} \to \mathcal{F}$ with Hilbert polynomial $P$, up to equivalence. According to Huybrechts and Lehn Proposition 2.2.7 (https://ncatlab.org/nlab/files/HuybrechtsLehn.pdf), we have the following result:
For all $s \in S$ and $q_{0} \in \pi^{-1}(s)$ corresponding to quotient $\mathcal{H}_{s} \to F$ with kernel $K$ as sheaves on $X_{s} = f^{-1}(s)$, we have the exact sequence $$ 0 \to \text{Hom}_{X_{s}}(K, F) \to T_{q_{0}}Q \xrightarrow[]{d\pi} T_{s}S \xrightarrow[]{\mathfrak{o}} \text{Ext}^{1}_{X_{s}}(K, F)$$ where $\mathfrak{o}$ is the obstruction map.
So my question is, what are the further entries in this exact sequence? Basically, I'm trying to compute the cokernel of the obstruction map $\mathfrak{o}$, but I'm struggling with how abstractly it is defined in Huybrechts and Lehn. Is there any geometrically intuitive way of understanding what the image or cokernel of an obstruction map encodes? Any references which might help with computing it?
