Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow X$ over $\mathcal{A}\times X$.

There exists a canonical connection $\mathbb{A}$ on $\mathbb{E}$ which is flat in the $\mathcal{A}$ direction and equal to $A$ on the slice $\{A\}\times X$. We obtain the following curvature: $$ \begin{cases} \mathbb{A}^2(v,w)= R_A(v,w) & \text{for $v,w\in T_xX$} \\ \mathbb{A}^2(\alpha,v)=\alpha(v) & \text{for $\alpha\in T_A\mathcal{A}$,$v\in T_xX$ } \\ \mathbb{A}^2(\alpha,\beta)=0 & \text{for $\alpha,\beta \in T_A\mathcal{A} $}. \end{cases} $$ These identities can be found in Donaldson's "INFINITE DETERMINANTS, STABLE BUNDLES AND CURVATURE" p236 and Itoh and Nakajima's "Yang-Mills Connections and Einstein-Hermitian Metrics" p451.

I do not understand the pairing of the middle line. How does it follow from the definition?