Here we consider the moduli space of Gieseker semistable sheaves on $X=\mathbb{P}^3$ (with respect to the ample divisor $H$) with Chern character $(0,0,L,n-2)$. We show that in this case, semistable=stable so that Joyce's sheaf-theoretic invariant agrees with the Behrend-Fantechi virtual fundamental class, using the trace-free obstruction theory of a complex in $D^b \operatorname{coh}(X)$. We compute the Behrend-Fantechi virtual fundamental class in the rational Chow ring of $\operatorname{Gr}(2,4)$, and construct Joyce's invariant in the Lie algebra of the homology of rigidified piecewise-linear higher stack of objects as mentioned in the previous talk, via pushforward.