For unbounded complexes, the claim is false. Consider the acyclic complex $\cdots \stackrel{2}{\to} \mathbb{Z}/4 \stackrel{2}{\to} \cdots$ of $\mathbb{Z}/4$-modules. This is a complex of projective modules (so, in the dual category, it's a complex of injective objects). But tensoring it with $\mathbb{Z}/2$, one obtains a complex which has cohomology in every degree. (This counterexample is in Gelfand/Manin.)

If $I^\bullet$ is an acyclic complex of injective objects, bounded below, it is contractible, i.e. homotopy equivalent to the zero complex. Since any additive functor (exact or not) has to preserve homotopy equivalences, it follows that in such a case $I^\bullet \otimes M$ is acyclic as well.

@Peter: Indeed, there is such a construction by Freyd and Adelman, see here and here. Martin: If you allow that $\mathrm{Hom}(T,A)$ is merely equivalent to $A$ (instead of being isomorphic), which I know you don't, then using the free abelian category generated by a single object as $T$ answers your question in the affirmative. Explicitly, it is given as $[\mathrm{Ab}_{fp},\mathrm{Ab}]_{fp}$.

This too is not answering the question, but note that -- if $\mathcal{O}_X$ is coherent (for instance, if $X$ is locally Noetherian) -- then $\mathrm{Coh}(X)$ is the smallest abelian subcategory of $\mathrm{Mod}(\mathcal{O}_X)$ containing the locally free $\mathcal{O}_X$-modules in an "indexed"/"fibered" sense, since any coherent sheaf is locally a cokernel of a map between finite locally free modules. (Coherence of $\mathcal{O}_X$ is needed to ensure that finite locally free modules are coherent.)

Sorry for bumping an old question. Can you specify what you mean with "generators and relations" in this context?

Sorry, I actually meant this article. "The purpose of this expository note is to describe duality and trace in a symmetric monoidal category, along with important properties (including naturality and functoriality), and to give as many examples as possible."

Keywords for the "self-eating" interpretation are the graphical calculus for tensor categories and string diagrams. Pointers are an article by Kate Ponto and Mike Shulman (see also accompanying slides) and a blog post by sigfpe.

The elementary characterization is also very useful in a topos-internal context. For example, a scheme $X$ is of dimension $\leq n$ if and only if, from the internal perspective of the little Zariski topos $\mathrm{Sh}(X)$, the (then plain old) ring $\mathcal{O}_X$ is of Krull dimension $\leq n$. See Proposition 3.13 of these sketchy notes.

@Heinrich: If I'm not mistaken, the expression you wrote is the group of Čech cochains, not yet the true $E_2$ term (which is a subquotient of your term). Also, the Stacks Project has $\prod$ instead of $\bigoplus$, but I don't know what's the best convention.

In your introductory paragraph, why do you need to assume that $X$ is sober? Sobriety seems superfluous to me.

Direct link to (just) the frontispiece.