3 Typos

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 $X \rightarrow [X, X]$.

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

$\mathit{nat}(\hom(-,$\mathit{nat}(\hom(-, 1), \hom(-, 1))$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, int^{B, C} F(B) \times G(C) \times \hom(-, B \otimes C)$$Moreover, Brian Day showed that \otimes_\mathbb{C} makes \mathbf{Set}^{\mathbb{C}^{op}} a monoidal (bi)closed category, with the Yoneda embedding y_\mathbb{C} \colon \mathbb{C} \rightarrow \mathbf{Set}^{\mathbb{C}^{op}} preserving the structure (i.e. not only does y_\mathbb{C} preserve tensors, but any existing linear exponents). This means that y_\mathbb{C} rises to the 2-functor Y \colon \mathbb{C}-\mathbf{Cat} \rightarrow \mathbf{Set}^{\mathbb{C}^{op}}-\mathbf{Cat}. By Yoneda, the underlying functor U \colon \mathbb{C}-\mathbf{Cat} \rightarrow \mathbf{Cat} factors through Y followed by the underlying functor V of \mathbf{Set}^{\mathbb{C}^{op}}-\mathbf{Cat}. Since the Yoneda functor y_\mathbb{C} also preserves equalisers, every monoid 1_X X in \mathbb{C} has a representation as a submonoid of y_\mathbb{C}(X)^{y_\mathbb{C}(X)} in \mathbf{Set}^{\mathbb{C}^{op}}. Then "\mathbf{Set}^{\mathbb{C}^{op}}, and X is a submonoid of [X, X] \in \mathbb{C} iff the linear exponent [X, X] exists in \mathbb{C}. "The second construction" is:$$V(E) \colon U(1_X) = V(1_{y_\mathbb{C}(X)}) \rightarrow V(1_{[y_\mathbb{C}(X), y_\mathbb{C}(X)]})$$All of the above should be quantified with "I think" :-) 2 Elaboration I will try to elaborate a bit more on the subject. Let us assume that \mathbb{C} is symmetric monoidal closed and has equalisers. Then any monoid X internal to \mathbb{C} admits embedding y_{1_X} \colon 1_X \rightarrow \mathbb{C}^{1_X^{op}}. The object of natural transformations \mathit{nat}(\hom(-, 1), \hom(-, 1)) by definition is the equaliser of l, r \colon [X, X] \rightarrow [X, [X, X]] in \mathbb{C}, where l is the transposition of:$$\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)$$Moreover, Brian Day showed that \otimes_\mathbb{C} makes \mathbf{Set}^{\mathbb{C}^{op}} a monoidal (bi)closed category, with the Yoneda embedding y_\mathbb{C} \colon \mathbb{C} \rightarrow \mathbf{Set}^{\mathbb{C}^{op}} preserving the structure (i.e. not only does y_\mathbb{C} preserve tensors, but any existing linear exponents). This means that y_\mathbb{C} rises to the 2-functor Y \colon \mathbb{C}-\mathbf{Cat} \rightarrow \mathbf{Set}^{\mathbb{C}^{op}}-\mathbf{Cat}. By Yoneda, the underlying functor U \colon \mathbb{C}-\mathbf{Cat} \rightarrow \mathbf{Cat} factors through Y followed by the underlying functor V of \mathbf{Set}^{\mathbb{C}^{op}}-\mathbf{Cat}. Since the Yoneda functor y_\mathbb{C} also preserves equalisers, every monoid 1_X in \mathbb{C} has a representation as y_\mathbb{C}(X)^{y_\mathbb{C}(X)} in \mathbf{Set}^{\mathbb{C}^{op}}. Then "the second construction" is:$$V(E) \colon V(1_{y_\mathbb{C}(X)}) \rightarrow V(1_{[y_\mathbb{C}(X), y_\mathbb{C}(X)]})$$All of the above should be quantified with "I think" :-) 1 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]\$.