## 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.