Let $F_\bullet$ be a free resolution of the module $M$ and let $G_\bullet$ be a free resolution of the module $N$. Consider the double complex $(A_{i, j}, d, \delta)$ defined as follows: \begin{enumerate} \item set $A_{i, j} = F_i \otimes_R G_j$, \item set $d_{i, j} : F_i \otimes_R G_j \to F_{i-1} \otimes G_j$ equal to $d_{F, i} \otimes \text{id}$, and \item set $\delta_{i, j} : F_i \otimes_R G_j \to F_i \otimes G_{j-1}$ equal to $\text{id} \otimes d_{G, j}$. \end{enumerate} This double complex is usually simply denoted $F_\bullet \otimes_R G_\bullet$. \medskip\noindent Since each $G_j$ is free, and hence flat we see that each row of the double complex is exact except in homological degree $0$. Since each $F_i$ is free and hence flat we see that each column of the double complex is exact except in homological degree $0$. Hence the double complex satisfies the conditions of Lemma \ref{lemma-no-spectral-sequence}. \medskip\noindent To see what the lemma says we compute $R(A)_\bullet$ and $U(A)_\bullet$. Namely, \begin{eqnarray*} R(A)_i & = & \Coker(A_{1, i} \to A_{0, i}) \\ & = & \Coker(F_1 \otimes_R G_i \to F_0 \otimes_R G_i) \\ & = & \Coker(F_1 \to F_0) \otimes_R G_i \\ & = & M \otimes_R G_i \end{eqnarray*} In fact these isomorphisms are compatible with the differentials $\delta$ and we see that $R(A)_\bullet = M \otimes_R G_\bullet$ as homological complexes. In exactly the same way we see that $U(A)_\bullet = F_\bullet \otimes_R N$. We get \begin{eqnarray*} \text{Tor}_i^R(M, N) & = & H_i(F_\bullet \otimes_R N) \\ & = & H_i(U(A)_\bullet) \\ & = & H_i(R(A)_\bullet) \\ & = & H_i(M \otimes_R G_\bullet) \\ & = & H_i(G_\bullet \otimes_R M) \\ & = & \text{Tor}_i^R(N, M) \end{eqnarray*} Here the third equality is Lemma \ref{lemma-no-spectral-sequence}, and the fifth equality uses the isomorphism $V \otimes W = W \otimes V$ of the tensor product. \medskip\noindent Functoriality. Suppose that we have $R$-modules $M_\nu$, $N_\nu$, $\nu = 1, 2$. Let $\varphi : M_1 \to M_2$ and $\psi : N_1 \to N_2$ be morphisms of $R$-modules. Suppose that we have free resolutions $F_{\nu, \bullet}$ for $M_\nu$ and free resolutions $G_{\nu, \bullet}$ for $N_\nu$. By Lemma \ref{lemma-compare-resolutions} we may choose maps of complexes $\alpha : F_{1, \bullet} \to F_{2, \bullet}$ and $\beta : G_{1, \bullet} \to G_{2, \bullet}$ compatible with $\varphi$ and $\psi$. We claim that the pair $(\alpha, \beta)$ induces a morphism of double complexes $$ \alpha \otimes \beta : F_{1, \bullet} \otimes_R G_{1, \bullet} \longrightarrow F_{2, \bullet} \otimes_R G_{2, \bullet} $$ This is really a very straightforward check using the rule that $F_{1, i} \otimes_R G_{1, j} \to F_{2, i} \otimes_R G_{2, j}$ is given by $\alpha_i \otimes \beta_j$ where $\alpha_i$, resp.\ $\beta_j$ is the degree $i$, resp.\ $j$ component of $\alpha$, resp.\ $\beta$. The reader also readily verifies that the induced maps $R(F_{1, \bullet} \otimes_R G_{1, \bullet})_\bullet \to R(F_{2, \bullet} \otimes_R G_{2, \bullet})_\bullet$ agrees with the map $M_1 \otimes_R G_{1, \bullet} \to M_2 \otimes_R G_{2, \bullet}$ induced by $\varphi \otimes \beta$. Similarly for the map induced on the $U(-)_\bullet$ complexes. Thus the statement on functoriality follows from the statement on functoriality in Lemma \ref{lemma-no-spectral-sequence}.