I'm reading a paper in which the following construction appears:
Let $(\mathcal{C}, \otimes, I)$ be a (dagger) monoidal category, and $(X, \triangledown, \bot)$ be a monoid in this category.


Step 1. *Externalise* the monoid by turning it into a monoid $(Hom(I,X),\cdot,\bot)$ where for $a,b : I \rightarrow X$ define $a \cdot b = \triangledown \circ (a \otimes b) \circ \lambda\_I ^{-1}$.


Step 2. *Represent* every element of $Hom(I,X)$ as and element of $Hom(X,X)$ by defining


$Y : Hom(I,X) \rightarrow Hom(X,X)$


$Y a = \triangledown \circ (a \otimes id\_x) \circ \lambda\_X ^{-1}$


(where $\lambda\_X : X \otimes I \rightarrow X$ is the appropriate structure map)


The author claims this is an example of the Yoneda-embedding. I can almost see how this is the case as $Y a$ is similar to $(a\cdot)$.


My questions are essentially: is this a standard construction, where can I read more about it? Is it really an instance of the Yoneda-embedding?


Also the two steps seem to be similar to the construction of the Kleisli category of a monad. The Wikipedia article on the Kleisli category includes two different presentations. Step 1 resembles the [definition of the Kleisli category](http://en.wikipedia.org/wiki/Kleisli_category#Formal_definition) and Step 2 looks like the $^\star$ operator used in the [definition of Kleisli triples](http://en.wikipedia.org/wiki/Kleisli_category#Extension_operators_and_Kleisli_triples). Are these constructions instances of a more general one?


[Here's a link to the paper](http://arxiv.org/abs/0812.2266)



Externalising monoids using the Yoneda-embedding and relation to Kleisli categories
For any monoidal category $\mathbb{C}$ there exists the "underlying" monoidal functor $\hom(I, -) \colon \mathbb{C} \rightarrow \mathbf{Set}$. As is the idea of a monoidal functor, it preserves structures defined by monoidal operations. Particularly, $\hom(I, -)$ lifts to the "underlying" 2-functor from the 2-category of $\mathbb{C}$-enriched categories to the 2-category of ordinary (that is: $\mathbf{Set}$-enriched) categories $U \colon \mathbb{C}$-$\mathbf{Cat} \rightarrow \mathbf{Cat}$.


A monoid $X$ internal to $\mathbb{C}$ is precisely a $\mathbb{C}$-enriched category $1\_X$ having a single object $1$ and $\hom(1, 1) = X$. From this perspective, the first construction corresponds to taking the underlying category of $1\_X$.


If $\mathbb{C}$ is closed and has equalisers, then something much stronger then proposition 2.6 should be true (i.e. proposition 2.6 should hold internally to $\mathbb{C}$). The monoid $1\_X$ via Yoneda $y\_{1\_X} \colon 1\_X \rightarrow \mathbb{C}^{1\_X^{op}}$ embeds into the category of presheaves on $1\_X$. Now the enriched Yoneda lemma says that $X$ is isomorphic to the object of natural transformations $\mathit{nat}(\hom(-, 1), \hom(-, 1))$, which, by the definition of a natural transformation, is a regular subobject of $[X, X] \in \mathbb{C}$.


We should get the second construction by applying the underlying functor to $X \rightarrow [X, X]$.




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I will try to elaborate a bit more on the subject.


Let us assume that $\mathbb{C}$ is symmetric monoidal closed and has equalisers. Then any monoid $X$ internal to $\mathbb{C}$ admits embedding $y\_{1\_X} \colon 1\_X \rightarrow \mathbb{C}^{1\_X^{op}}$. The object of natural transformations


$$\mathit{nat}(\hom(-, 1), \hom(-, 1))$$
by definition is the equaliser of $l, r \colon [X, X] \rightarrow [X, [X, X]]$ in $\mathbb{C}$, where $l$ is the transposition of:
$$\mu\_\mathbb{C} \circ (\mathit{id}\_{[X, X]} \otimes \nabla^\*) \colon [X, X] \otimes X \rightarrow [X, X]$$
$r$ is the other transposition of:
$$\mu\_\mathbb{C} \circ (\nabla^\* \otimes \mathit{id}\_{[X, X]}) \colon X \otimes [X, X] \rightarrow [X, X]$$
$\nabla^\* \colon X \rightarrow [X, X]$ is the transposition of the monoidal multiplication $X \otimes X \rightarrow X$, and $\mu\_\mathbb{C} \colon [X, X] \otimes [X, X] \rightarrow [X, X]$ is the internalised composition from $\mathbb{C}$.


The enriched Yoneda lemma says that
$$\mathit{nat}(\hom(-, 1), \hom(-, 1)) \approx \hom(1, 1) = X$$
Therefore the "arrows part" of $hom(-, 1)$ --- $e \colon X \rightarrow [X, X]$ is the equaliser of $l, r \colon [X, X] \rightarrow [X, [X, X]]$. Furthermore, because $hom(-, 1)$ is a functor, it maps the composition in $1\_X^{op}$ to the composition in $\mathbb{C}$, turning $e$ into a functor between internal monoids $E \colon 1\_X \rightarrow 1\_{[X, X]}$.


The second construction is given by the application of the underlying functor $U$ to $E$:


$$U(E) \colon U(1\_X) \rightarrow U(1\_{[X, X]})$$


One may perhaps use the weak version of the Yoneda lemma to construct $U(E)$ in case $\mathbb{C}$ is not monoidal closed with equalisers. However, there is also a more natural solution.


Let us recall that if $\mathbb{C}$ is monoidal, then its category of presheaves $\mathbf{Set}^{\mathbb{C}^{op}}$ inherits the monoidal structure via the very special case of convolution:


$$F \otimes\_\mathbb{C} G = \int^{B, C} F(B) \times G(C) \times \hom(-, B \otimes C)$$ 


Moreover, Brian Day showed that $\otimes\_\mathbb{C}$ makes $\mathbf{Set}^{\mathbb{C}^{op}}$ a monoidal (bi)closed category, with the Yoneda embedding $y\_\mathbb{C} \colon \mathbb{C} \rightarrow \mathbf{Set}^{\mathbb{C}^{op}}$ preserving the structure (i.e. not only does $y\_\mathbb{C}$ preserve tensors, but any existing linear exponents). This means that $y\_\mathbb{C}$ rises to the 2-functor $Y \colon \mathbb{C}$-$\mathbf{Cat} \rightarrow \mathbf{Set}^{\mathbb{C}^{op}}$-$\mathbf{Cat}$. By Yoneda, the underlying functor $U \colon \mathbb{C}$-$\mathbf{Cat} \rightarrow \mathbf{Cat}$ factors through $Y$ followed by the underlying functor $V$ of $\mathbf{Set}^{\mathbb{C}^{op}}$-$\mathbf{Cat}$.


Since the Yoneda functor $y\_\mathbb{C}$ also preserves equalisers, every monoid $X$ in $\mathbb{C}$ has a representation as a submonoid of $y\_\mathbb{C}(X)^{y\_\mathbb{C}(X)}$ in $\mathbf{Set}^{\mathbb{C}^{op}}$, and $X$ is a submonoid of $[X, X] \in \mathbb{C}$ iff the linear exponent $[X, X]$ exists in $\mathbb{C}$. "The second construction" is:


$$V(E) \colon U(1\_X) = V(1\_{y\_\mathbb{C}(X)}) \rightarrow V(1\_{[y\_\mathbb{C}(X), y\_\mathbb{C}(X)]})$$


