Definitions of the Atiyah class

Question

Let $E$ be a coherent sheaf on $X$, then its Atiyah class is defined as follows: There is a map $\alpha: \mathcal{O}_{\Delta X}\rightarrow \Delta_*\Omega^1_X$ where $\Delta:X \rightarrow X\times X$ denotes the diagonal embedding. Now apply $p_*(\_\otimes q^*E)$ for the canonical projections $p,q: X \times X\rightarrow X$ to get a map $p_*(\alpha \otimes 1): E\rightarrow E\otimes \Omega_X^1$ (here note that $p_*(\mathcal{O}_{\Delta X}\otimes q^*E)\cong E$). Sometimes this map is written as $Hom(E,E) \rightarrow \Omega_X^1[1]$, so my question is: What is this map now? Is it something like $f\mapsto tr(f)\cdot \alpha...$? thanks