Let $A,B$ be $k$-linear (possibly, dg-)categories, let $f,g:Ato B$ be two linear functors, and let $T:fRightarrow g$ be a natural isomorphism.

If one denotes by $C_bullet(A,A)$ the standard Hochschild chain complex of $A$ with coefficients in itself, then $f$ and $g$ induce two chain maps $mathbf{f},mathbf{g}:C_bullet(A,A)to C_bullet(B,B)$.

One knows from general facts about the homotopy theory of dg-categories that there exists a homotopy $mathbf{T}$ between $mathbf{f}$ and $mathbf{g}$.

**Here is my question**: Is there a reference where one can find an explicit expression for this homotopy?

*Ideally, I’d like a reference where the compatibility with the mixed structure on Hochschild chains (that is, with Connes’ boundary operator) is discussed.*

One can of course guess (and, with some work, prove) a formula. I’m almost sure such a formula is already written somewhere, but I couldn’t find a reference.

*Note that I’d already be happy to know about a reference for (dg-)algebras (in which case a natural isomorphism $T:fRightarrow g$ is just the data of an invertible element $b$ of $B$ such that for every $ain A$, $bf(a)=g(a)b$).*