Transferred maps are functorial up to homotopy, with strict naturality limits

Transfer of maps and homotopies. The composite $\bar{f}=p_{Y}f{\mathrm{ı}}_X$ of cochain maps is a cochain map, so $d\bar{f}=\bar{f}d$ by [L3]. If $f-g=ds+sd$, then $\bar{f}-\bar{g}=p_Y(f-g){\mathrm{ı}}_X=p_{Y}ds{\mathrm{ı}}_X+p_{Y}sd{\mathrm{ı}}_X=d(p_{Y}s{\mathrm{ı}}_X)+(p_{Y}s{\mathrm{ı}}_X)d$, using $p_{Y}d=d{p}_Y$ and $d{\mathrm{ı}}_X={\mathrm{ı}}_{X}d$; hence $\bar{s}=p_{Y}s{\mathrm{ı}}_X$ is a degree-$(-1)$ homotopy $\bar{f}\simeq \bar{g}$. Therefore homotopic maps have homotopic transfers, and transfer is well defined on homotopy classes.

Functoriality up to homotopy. The identity transfers strictly: $\overline{1_{X^{\bullet }}}=p_X{1}_{X^{\bullet }}{\mathrm{ı}}_X=p_X{\mathrm{ı}}_X=1_{{\bar{X}}^{\bullet }}$. For composable $f,g$ set $k:=p_{Z}g{h}_{Y}f{\mathrm{ı}}_X$; then $\bar{g}\bar{f}=p_{Z}g{\mathrm{ı}}_Y{p}_{Y}f{\mathrm{ı}}_X=p_{Z}g(1_{Y^{\bullet }}-(d{h}_Y+h_{Y}d))f{\mathrm{ı}}_X=\overline{(gf)}-(p_{Z}gd{h}_{Y}f{\mathrm{ı}}_X+p_{Z}g{h}_{Y}df{\mathrm{ı}}_X)$, and since $p_Z,g,f,{\mathrm{ı}}_X$ are cochain maps the two correction terms are $d(p_{Z}g{h}_{Y}f{\mathrm{ı}}_X)$ and $(p_{Z}g{h}_{Y}f{\mathrm{ı}}_X)d$, that is $dk$ and $kd$. Hence $\bar{g}\bar{f}=\overline{(gf)}-(dk+kd)$, equivalently $\overline{(gf)}-\bar{g}\bar{f}=dk+kd$, so $\overline{(gf)}\simeq \bar{g}\bar{f}$ with this sign convention.

Strictness fails. Take $\mathcal{A}$ the category of vector spaces over a field, let $K$ be the two-term complex $k\stackrel{1}{\to }k$ concentrated in degrees $0,1$ with zero neighbouring terms, let ${\bar{X}}^{\bullet }$ be a second copy of it and $X^{\bullet }={\bar{X}}^{\bullet }\oplus K$, and use the split-off retract of [L2] with $\mathrm{ı}$ the inclusion of the first summand, $p$ the projection onto it and $h^1=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)$, $h^0=h^2=0$. In each degree let $f=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$ and $g=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ in the coordinates ${\bar{X}}^0\oplus K^0$ respectively ${\bar{X}}^1\oplus K^1$; the components in degrees $0$ and $1$ agree, so both maps commute with the only nonzero differential $d^0=1$, so $f$ and $g$ are cochain maps. Then $gf=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)=\mathrm{ı}p$, and the transfers are $\bar{f}=pf\mathrm{ı}=0$ and $\bar{g}=pg\mathrm{ı}=0$ because $f\mathrm{ı}$ and $g\mathrm{ı}$ land in the complementary summand killed by $p$, while $\overline{(gf)}=p(gf)\mathrm{ı}=p\mathrm{ı}=1_{{\bar{X}}^{\bullet }}$. Hence $\overline{(gf)}-\bar{g}\bar{f}=1_{{\bar{X}}^{\bullet }}\mathrm{\ne }0$, so transfer is not a strict functor on cochain maps.

Strict naturality for commuting morphisms. Let $f:X^{\bullet }\to Y^{\bullet }$ and $g:Y^{\bullet }\to Z^{\bullet }$ satisfy $f{\mathrm{ı}}_X={\mathrm{ı}}_Y\bar{f}$, $p_{Y}f=\bar{f}{p}_X$, $g{\mathrm{ı}}_Y={\mathrm{ı}}_Z\bar{g}$ and $p_{Z}g=\bar{g}{p}_Y$. Then $\overline{(gf)}=p_{Z}gf{\mathrm{ı}}_X=p_{Z}g{\mathrm{ı}}_Y\bar{f}=\bar{g}{p}_Y{\mathrm{ı}}_Y\bar{f}=\bar{g}\bar{f}$, the last step by [L1]; under these hypotheses the transfer $p_{Y}f{\mathrm{ı}}_X=\bar{f}{p}_X{\mathrm{ı}}_X=\bar{f}$ is the given $\bar{f}$, so the computation compares the transfer of the composite with the composite of the transfers. In this situation identity and composition are preserved strictly, not merely up to homotopy.

Conclusion. Step 1.1 shows that transfer sends cochain maps to cochain maps and homotopic maps to homotopic maps, so it is well defined on homotopy classes of cochain maps; step 1.2 shows that it preserves identities strictly and composition up to the explicit homotopy $p_{Z}g{h}_{Y}f{\mathrm{ı}}_X$, so it is functorial on homotopy classes while not being a strict functor on cochain maps; step 1.3 exhibits cochain maps with $\bar{f}=\bar{g}=0$ and $\overline{(gf)}=1$, which establishes that failure; and step 1.4 gives strict functoriality on the subcategory of morphisms commuting with the chosen retract data. Since the transfer uses the chosen projections and inclusions, while the displayed comparison homotopy also uses the chosen homotopies, clause 4 records that no choice-free, canonical or confluent transfer and no independence of the chosen data is being claimed. ∎