This is a simple question about Tor and Ext functors.

Let $R$ be a commutative ring, and let $0 \to M' \to M \to M'' \to 0$ and $0 \to N' \to N \to N'' \to 0$ be short exact sequences of $R$-modules. Then using the connecting homomorphisms from the long exact sequences, we can build the following diagram:

\begin{matrix} \operatorname{Tor}^R_i(M'',N'') & \longrightarrow & \operatorname{Tor}^R_{i-1}(M',N'') \\ \downarrow && \downarrow \\ \operatorname{Tor}^R_{i-1}(M'',N') & \longrightarrow & \operatorname{Tor}^R_{i-2}(M',N') \end{matrix}

Is this diagram commutative? A similar question can be asked about the Ext functor.

I think the answer might depend on the definitions used: there are a couple of "equivalent" definitions for Tor/Ext. My motivation is actually to understand what equivalent should really mean here. (E.g. derived functors can be defined up to isomorphisms of $\delta$-functors, however $\operatorname{Tor}(\cdot, \cdot)$ seems to be something like a "2-variable $\delta$-functor".)