$\def\A{\mathcal{A}}
\def\J{\mathcal{J}}$I am reading P. Seidel, *Fukaya Categories and Picard-Lefschetz Theory*, and in (1l) he defines the *Yoneda embedding* of a (non-unital) A$_\infty$-category $\A$ to be a certain (non-unital) A$_\infty$-functor $\mathcal{J}:\A\to\operatorname{nu-mod}(\A)$ (where $\operatorname{nu-mod}(\A)$ is the A$_\infty$-category of non-unital $\A$-modules).

I will copy here the definition of $\mathcal{J}$ as explained in *Dirichlet Branes and Mirror Symmetry*, 8.3.4, p. 612 (the notation has been adapted to Seidel's one):

We can now define something known as the Yoneda embedding, an A$_{\infty}$-functor $\J:\mathcal{A} \rightarrow \operatorname{nu-mod}(\mathcal{A})$. This has a simple definition, namely for $Y \in \operatorname{Ob}(\mathcal{A})$, we set $\J(Y)$ to be the module $X \mapsto \operatorname{Hom}_{\mathcal{A}}(X, Y)$. Of course, the latter is already a chain complex, coming from $\mu^1_{\mathcal{A}}$, and to define the module structure, we need to provide maps as in (8.9). However, as in this case $\mathcal{M}(X_{d-1})=\operatorname{Hom}_{\mathcal{A}}(X_{d-1}, Y)$ and $\mathcal{M}\left(X_0\right)=\operatorname{Hom}_{\mathcal{A}}\left(X_0, Y\right)$, these maps are just given by $\mu^d_{\mathcal{A}}$. So this gives the functor Yon as a map from objects of $\mathcal{A}$ to objects of $\operatorname{nu-mod}(\mathcal{A})$; however, we have to define the higher maps, $$\J^d: \operatorname{Hom}_{\mathcal{A}}(X_{d-1}, X_d) \otimes \cdots \otimes \operatorname{Hom}_{\mathcal{A}}(X_0, X_1) \rightarrow\\ \operatorname{Hom}_{\operatorname{nu-mod}(\mathcal{A})}\left(\J\left(X_0\right), \J\left(X_d\right)\right)[1-d] \text {. } $$

So given $a_1, \ldots, a_d$, we need to give a pre-homomorphism, i.e., a collection of maps $$ \begin{gathered} t^n: \J(X_0)(Y_{n-1}) \otimes \operatorname{Hom}_{\mathcal{A}}(Y_{n-2}, Y_{n-1}) \otimes \cdots \otimes \operatorname{Hom}_{\mathcal{A}}(Y_0, Y_1) \rightarrow \\ \J\left(X_d\right)\left(Y_0\right)\left[2-d-n+\sum |a_i|\right], \end{gathered} $$ i.e., $$ \begin{aligned} (\J^d)^n: \operatorname{Hom}_{\mathcal{A}}\left(Y_{n-1}, X_0\right) & \otimes \operatorname{Hom}_{\mathcal{A}}\left(Y_{n-2}, Y_{n-1}\right) \otimes \cdots \otimes \operatorname{Hom}_{\mathcal{A}}\left(Y_0, Y_1\right) \rightarrow \\ & \operatorname{Hom}_{\mathcal{A}}(Y_0, X_d)\left[2-d-n+\sum |a_i|\right]. \end{aligned} $$

There is an obvious choice, i.e., $$ \left(\J^d\left(a_d, \ldots, a_1\right)\right)^n(b, b_{n-1}, \ldots, b_1)=\mu^{d+n}_{\mathcal{A}}(a_d, \ldots, a_1, b, b_{n-1}, \ldots, b_1). $$

My questions are:

Why is the Yoneda embedding an 'embedding'?

Is there a definition of an 'embedding of A$_\infty$-categories'? If so, does the Yoneda embedding satisfy this definition?

I guess an answer for 2 counts as an answer for 1.

An A$_\infty$-category $\A$ is said to be *c-unital* if $H(\A)$ is a unital category. A non-unital A$_\infty$-functor between non-unital A$_\infty$-categories $\mathcal{F}:\mathcal{A}\to\mathcal{B}$ is said to be *cohomologically full and faithful* if $H(\mathcal{F}):H(\mathcal{A})\to H(\mathcal{B})$ is full and faithful (Seidel, (1b)). It turns out that $\J$ is cohomologically full and faithful whenever $\A$ is c-unital (Seidel, Corollary 2.13). Is this the sense in which $\J$ is an embedding? (So maybe a more appropriate name for $\J$ would be the “Yoneda cohomological embedding.”)