I have a question about certain detail in following answer by Denis Nardin adressing the concept of presheaves with transfer (mostly known in constructions in motivic homotopy theory) from viewpoint of classical stable homotopy theory.

The rough summary is that the notion of a "transfer" appears already (... or phrased differently has it's counterpart) in classical SHT in guise of so called Becker-Gottlieb transfer which to a map $f: X \to Y$ of top spaces (**#Correction:** (thanks to John Rognes' remark) BG is not defined for arbitrary $f$; but let assume for those maps and their compositions we are considering in this question in is assumed to be defined)

associates in category $\text{Sp}$ of spectra certain map
$f^*: \Sigma^\infty_+Y\to \Sigma^\infty_+X$ for which for composition $X\xrightarrow{f}Y\xrightarrow{g} Z$ the equation $g^*f^*=(gf)^*$ is literally an equality only after passing to homotopy category (...and so "rectifiyng" the functoriality). In other words, one may think of this equality (...in spirit of $\infty$-categorical philosophy) as a rather compliciated datum of a homotopy between the two maps (then with coherent homotopies of homotopies on the "next" layer, and so on ...); at all a rather complicated datum.

But instead of working directly with spectra (which may be regarded as space-valued homology functors $\mathrm{Top}\to \mathrm{Top}_*$ & some consistence conditions), one could also study functors $\mathrm{Cor}\to \mathrm{Top}_* $ (from the category $ \mathrm{Cor}$ of correspondences & some consistence conditions; see the linked discussion for details), which turns of to be given as $D(\mathbb{Z})$, the derived cat of $\mathbb{Z}$ endowed with natural forgetful functor $D(\mathbb{Z})\to \mathrm{Sp}$ to the cat of spectra identifying $D(\mathbb{Z})$ with spectra carying additionally $H\mathbb{Z}$-module structure.

The pun seems to be that if we restrict to the coherence conditions for compositions of Becker-Gottlieb transfers given above to those spectra inside $D(\mathbb{Z})$, then according to the statement in the linked answer these "hold on the nose".

**Question:** And exactly this point I not completely understand. What does here mean that these coherences conditions on the transfers "hold on the nose" if we restrict ourself to $D(\mathbb{Z})$.

In the answer is stated that the mapping spaces to $D(\mathbb{Z})$ is given dy $X \mapsto C_*(X)$ (ie associating ato $X$ it's complex and pass to derived cat of complexes).

The latter formulation suggests that working inside $D(\mathbb{Z})$ makes the coherences conditions on the composition of Becker-Gottlieb transfers in certain way more "amenable"/ "nicer"/ better behaved/controllable in certain sense (?) justifying the formulation that in $D(\mathbb{Z})$ these hold more or less "on the nose" in contrast to the "messy" datum if we work in $\text{Sp}$ instead.

What are the advantages of this viewpoint and in which sense in $D(\mathbb{Z})$ the coherence relations hold more or less "on the nose"? Does this mean that in $D(\mathbb{Z})$ these can be written down explicitly , while in $\text{Sp}$ they exist only abstractly?

1more comment