Natural transformations of A∞A_\infty-functors (between dg-categories) are “directed homotopies” (reference?)

Let A and B be dg-categories over a field, viewed as A-categories. The A-category (actually, dg-category) of strictly unital A-functors AB will be denoted by Fun(A,B). It is described explicitly (in the non-unital case) for example in P. Seidel’s book (“Fukaya category and Picard-Lefschetz theory”). Let Δ1 be the 1-simplex category, namely, the linear category freely generated over the diagram 01. Call Mor(B)=Fun(Δ1,B). It can be described explicitly as the dg-category of morphisms in B, see for example “Internal Homs via extensions of dg functors” by Canonaco and Stellari, after Remark 2.9. There are natural source and target dg-functors S,T:Mor(B)B.

Now, I would like to prove that closed degree 0 morphisms FG of Fun(A,B) are in bijection with A-functors φ:AMor(B) such that Sφ=F and Tφ=G. That is, morphisms of A-functors are represented by “directed homotopies”. I think that this result is quite tautological if one employs the concrete definitions contained in Seidel’s book; the real pain is called sign conventions. If someone knows a conceptual argument which proves this, or – even better – has a reference somewhere, it would be of great help. Thanks in advance!

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Source : Link , Question Author : Francesco Genovese , Answer Author : Community

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