I am studying enriched categories, and as I wrote in my previous question https://mathoverflow.net/questions/404288/how-is-the-morphism-of-composition-in-the-enriched-category-of-modules-construct, this is very  difficult because there are no elementary books on this topic (so I hope that specialists in category theory will not kill me for what I am asking here).

In particular, I am looking from time to time at the Tannaka duality theorem for modules over monoids in a symmetric monoidal category, and I have a feeling that it can be proved without the [Yoneda lemma which is suggested as an instrument in this situation][2].  

Am I right?

My impression is that everything must be simple, although I can't restore the details (and I hope that somebody will help me).

Suppose we have a monoid $A$ in a symmetric monoidal category $V$ and we consider the enriched category $_AV$ of modules over $A$, and the forgetful functor $F:{_AV}\to V$. It's easy to see that the family of maps 
$$
\alpha_X:A\to[F(X),F(X)],
$$ 
where $\alpha_X$ corresponds to the $A$-module structure $\mu_X:A\otimes X\to X$ on $X\in\operatorname{Ob}({_AV})$, is what is called a *wedge* from $F$ to $F$.  

Our aim is to prove that for any other wedge 
$$
\beta_X:B\to[F(X),F(X)]
$$ 
there is a unique morphism $\omega:B\to A$ such that the diagrams in the family 
[![universal property of wedges][3]][3]

(with $X\in\operatorname{Ob}({_AV})$) are commutative.

I guess that this $\omega$ is defined as the composition
$$
B\xrightarrow{\beta_A}[A,A] \xrightarrow{[\iota,1_A]}[I,A]
\xrightarrow{\rho_{[\iota,1_A]}^{-1}}[I,A]\otimes I
\xrightarrow{\operatorname{ev}^A_I}A
$$
(where $I$ is the unit in $V$, $\iota:I\to A$  the unit in the monoid $A$ and $\rho_{[\iota,1_A]}$ the right identity for $[\iota,1_A]$), but I don't understand how to prove the commutativity of diagrams (1).

In particular, the special case of (1)
[![universal property of wedges with X = A][4]][4]
 
is also a puzzle for me. This is the same as the equality of the morphisms
$$
B\otimes A\xrightarrow{\beta_A\otimes 1_A}[A,A]\otimes A \xrightarrow{[\iota,1_A]\otimes 1_A}[I,A]\otimes A
\xrightarrow{\rho_{[\iota,1_A]}^{-1}\otimes 1_A}[I,A]\otimes I\otimes A
\xrightarrow{\operatorname{ev}^A_I\otimes 1_A}A\otimes A \xrightarrow{\mu_A}A
$$
and
$$
B\otimes A\xrightarrow{\beta_A\otimes 1_A}[A,A]\otimes A \xrightarrow{\operatorname{ev}^A_A}A
$$
but why are they equal? I can check this only in the situation when $A$ is a usual monoid (in the category of sets) or a usual algebra.

I would appreciate if anybody could show this "bookkeeping" (as Fosco Loregian calls it), because for an outsider like me this is  incomprehensible. As I wrote in my previous question, each elementary step in this field is a problem, it requires weeks of pondering for me.

P.S. The applications of the Yoneda lemma are a gap in my education. If some kind person explains to me how it works, I will transfer this esoteric knowledge to the Moscow Mathematical School (with reference to this benefactor).

  [1]: https://mathoverflow.net/questions/404288/how-is-the-morphism-of-composition-in-the-enriched-category-of-modules-construct
  [2]: https://ncatlab.org/nlab/show/Tannaka+duality#ForVModules
  [3]: https://i.sstatic.net/N50C0.png
  [4]: https://i.sstatic.net/Mr3Po.png