Let $\mathbf A$ and $\mathbf B$ be dg-categories over a field, viewed as $A_\infty$-categories. The $A_\infty$-category (actually, dg-category) of strictly unital $A_\infty$-functors $\mathbf A \to \mathbf B$ will be denoted by $\mathrm{Fun}_\infty(\mathbf A, \mathbf B)$. It is described explicitly (in the non-unital case) for example in P. Seidel's book ("Fukaya category and Picard-Lefschetz theory"). Let $\Delta^1$ be the $1$-simplex category, namely, the linear category freely generated over the diagram $0 \to 1$. Call $\mathrm{Mor}(\mathbf B) = \mathrm{Fun}_\infty(\Delta^1, \mathbf B)$. It can be described explicitly as the dg-category of morphisms in $\mathbf B$, 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 $S, T \colon \mathrm{Mor}(\mathbf{B}) \to \mathbf B$.

Now, I would like to prove that closed degree $0$ morphisms $F \to G$ of $\mathrm{Fun}_\infty(\mathbf A, \mathbf B)$ are in bijection with $A_\infty$-functors $\varphi \colon \mathbf A \to \mathrm{Mor}(\mathbf B)$ such that $S \circ \varphi = F$ and $T \circ \varphi = G$. That is, morphisms of $A_\infty$-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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