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Exercise (Universal Property of the Tensor Product). Let $(C_{\ast }, \partial )$, $(D_{\ast }, \partial )$, and $(E_{\ast }, \partial )$ be chain complexes. We will say that a collection of bilinear maps

\[ \{ f_{m,n}: C_{m} \times D_{n} \rightarrow E_{m+n} \} _{m,n \in \operatorname{\mathbf{Z}}} \]

satisfies the Leibniz rule if, for every pair of elements $x \in C_ m$ and $y \in D_{n}$, the identity

\[ \partial f_{m,n}(x,y)= f_{m-1,n}( \partial x, y ) + (-1)^{m} f_{m,n-1}( x, \partial y ) \]

holds in the abelian group $E_{m+n-1}$. Show that there is a canonical bijection from the collection of chain maps $f: C_{\ast } \boxtimes D_{\ast } \rightarrow E_{\ast }$ to the collection of systems of bilinear maps $\{ f_{m,n}: C_ m \times D_{n} \rightarrow E_{m+n} \} _{m,n \in \operatorname{\mathbf{Z}}}$ satisfying the Leibniz rule, given by the construction $f_{m,n}(x,y) = f( x \boxtimes y)$.