Let A and B be dg-categories over a field, viewed as A∞-categories. The A∞-category (actually, dg-category) of

strictly unitalA∞-functors A→B 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 0→1. 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 F→G of Fun∞(A,B) are in bijection with A∞-functors φ:A→Mor(B) such that S∘φ=F and T∘φ=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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**Attribution***Source : Link , Question Author : Francesco Genovese , Answer Author : Community*