Defining a graded ring structure on $\bigoplus_{i} \text{Ext}^i (A,A)$

I read that, using the fact that $ \text{Tot}^{\prod}(\text{Hom}(P_{\bullet} ,Q_{\bullet}))$ can by used to compute $ \text{Ext}^* (A,B)$ (which I understand), we can give $ \bigoplus_{i} \text{Ext}^i (A,A)$ the structure of a graded ring, considering elements of $ \text{Ext}^i (A,A)$ as chain maps $ P_{\bullet}\rightarrow Q_{\bullet}[i]$ modulo homotopy (where $ [i]$ denotes a shift by $ i$ places) and the ring structure is given by composing these. Apparently this follows as, using the total chain complex, we get that composition of homomorphisms $ \circ :\text{Hom}(A,B)\otimes \text{Hom}(B,C)\rightarrow \text{Hom}(A,C)$ induces a well-defined map $ \text{Ext}^i (A,B)\otimes \text{Ext}^j (B,C)\rightarrow \text{Ext}^{i+j} (A,C)$ and as we can identify cycles in $ \text{Tot}^{\prod}(\text{Hom}(P_{\bullet} ,Q_{\bullet}))$ with these maps.

Could anyone please explain to me how this identification works and why we can identify $ \text{Ext}^i(A,A)$ with the maps in this way.