Various books and many online notes and other posts such as Proving that $\operatorname{Tor}_n^R$ is a bifunctor give a rather touch-and-go treatment to Tor being a covariant functor. For example, in Matsumara's Commutative Ring Theory, Appendix B, the treatment for Tor is very brief with only the outline of how Tor is constructed. I am currently stuck on the claim that Tor is a covariant functor.
The following definition is from Dummit and Foote's Abstract Algebra that I am working with (commutative rings throughout).
If $\mathcal{C}$ and $\mathcal{D}$ are categories, we say that $\mathcal{F}$ is a covariant functor from $\mathcal{C}$ to $\mathcal{D}$ if:
(1) for every object $A$ of $\mathcal{C}$, $\mathcal{F}A$ is an object in $\mathcal{D}$
(2) for every morphism $f\in Mor_{\mathcal{C}}(A,B)$, we have $\mathcal{F}f\in Mor_{\mathcal{C}}(\mathcal{F}A,\mathcal{F}B)$
such that $\mathcal{F}(1_{A})=1_{\mathcal{F}A}$ and if $g\circ f$ is a composition of morphisms in $\mathcal{C}$, then $\mathcal{F}(gf)=\mathcal{F}(g)\mathcal{F}(f)$.
The definition of Tor that I am working with is from Matsumara's Commutative Ring Theory Appendix B:
In brief, the construction starts with two projective resolutions of $A$-modules $M$ and $N$, given by $P_{\bullet}$ and $Q_{\bullet}$ respectively, which after taking $M\otimes_{A}-$ and $-\otimes_{A}N$ respectively, we get two complexes $P_{\bullet}\otimes_{A}N$ and $M\otimes_{A}Q_{\bullet}$. Then we define $\mathrm{Tor}_{n}^{A}(M,N)\cong H_{n}(M\otimes_{A}Q_{\bullet})\cong H_{n}(P_{\bullet}\otimes_{A}N)$, where $H_{n}(K_{\bullet})$ is the $n$-th homology of the complex $K_{\bullet}$.
Most online notes are brief and simply outlines why Tor is a covariant functor. However I am having trouble figuring out the respective maps to show that Tor is indeed a covariant functor, getting lost in the many commutative diagrams.
My knowledge on homological algebra is highly elementary and introductory. However I would appreciate any hint or help.