# reference request – Explicit homotopy for Hochschild chains from natural isomorphism

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 $$ain A$$, $$bf(a)=g(a)b$$).