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It's been a long time since I tried to understand the deep meaning of the "Kan construction", or "nerve-realization" adjunction $$ \text{Lan}_y F \dashv N_F = \hom(F,1) $$ that exists among the left Kan extension of $F\colon \mathcal{A}\to \mathbf{D}$ ($\cal A$ small, $\bf D$ cocomplete) along the Yoneda embedding $y\colon {\cal A} \to \hat{\cal A}$. It can be expressed as $$ \text{Lan}_y F \dashv \text{Lan}_F y, $$ and this property seems pretty peculiar; especially if I think about the definition of a "Yoneda structure"[here, several comments in the discussion are mine].

1. Is there a reason why this is true?
2. What are other examples of a span of functors ${\bf C} \xleftarrow{G} {\cal A} \xrightarrow{F} {\bf B}$ such that $\text{Lan}_GF\dashv \text{Lan}_FG$?
3. My sensation is that this question acquires a (more?) meaning plunging the 2-category $\bf Cat$ in $\bf Prof$ in the usual way. The functor $N_F = \hom(F,1)$ is the image of $F$ via the canonical 2-functor $\varphi^{(-)} : {\bf Cat}^\text{co} \to \bf Prof$, and $N_F$ has a right adjoint $\hom(1,F)$; the Kan construction amounts to say that this extends to a triple of adjoints $$ \text{Lan}_yF = \varphi_F^! \dashv \varphi^F\dashv \varphi_F. $$ What's the meaning of this extension, and its universal property, in $\bf Prof$?

It's been a long time since I tried to understand the deep meaning of the "Kan construction", or "nerve-realization" adjunction $$ \text{Lan}_y F \dashv N_F = \hom(F,1) $$ that exists among the left Kan extension of $F\colon \mathcal{A}\to \mathbf{D}$ ($\cal A$ small, $\bf D$ cocomplete) along the Yoneda embedding $y\colon {\cal A} \to \hat{\cal A}$. It can be expressed as $$ \text{Lan}_y F \dashv \text{Lan}_F y, $$ and this property seems pretty peculiar; especially if I think about the definition of a "Yoneda structure"[here, several comments in the discussion are mine].

1. Is there a reason why this is true?
2. What are other examples of a span of functors ${\bf C} \xleftarrow{G} {\cal A} \xrightarrow{F} {\bf B}$ such that $\text{Lan}_GF\dashv \text{Lan}_FG$?
3. My sensation is that this question acquires a (more?) meaning plunging the 2-category $\bf Cat$ in $\bf Prof$ in the usual way. The functor $N_F = \hom(F,1)$ is the image of $F$ via the canonical 2-functor $\varphi^{(-)} : {\bf Cat}^\text{co} \to \bf Prof$, and $N_F$ has a right adjoint $\hom(1,F)$; the Kan construction amounts to say that this extends to a triple of adjoints $$ \text{Lan}_yF = \varphi_F^! \dashv \varphi^F\dashv \varphi_F. $$ What's the meaning of this extension, and its universal property, in $\bf Prof$?

It's been a long time since I tried to understand the deep meaning "Kan construction", o "nerve-realization" adjunction $$ \text{Lan}_y F \dashv N_F = \hom(F,1) $$ that exists among the left Kan extension of $F\colon \mathcal{A}\to \mathbf{D}$ ($\cal A$ small, $\bf D$ cocomplete) along the Yoneda embedding $y\colon {\cal A} \to \hat{\cal A}$. It can be expressed as $$ \text{Lan}_y F \dashv \text{Lan}_F y, $$ and this property seems pretty peculiar; especially if I think about the definition of a "Yoneda structure"[here, several comments in the discussion are mine].

1. Is there a reason why this is true?
2. What are other examples of a span of functors ${\bf C} \xleftarrow{G} {\cal A} \xrightarrow{F} {\bf B}$ such that $\text{Lan}_GF\dashv \text{Lan}_FG$?
3. My sensation is that this question acquires a (more?) meaning plunging the 2-category $\bf Cat$ in $\bf Prof$ in the usual way. The functor $N_F = \hom(F,1)$ is the image of $F$ via the canonical 2-functor $\varphi^{(-)} : {\bf Cat}^\text{co} \to \bf Prof$, and $N_F$ has a right adjoint $\hom(1,F)$; the Kan construction amounts to say that this extends to a triple of adjoints $$ Lan_yF = \varphi_F^! \dashv \varphi^F\dashv \varphi_F $$ what's the meaning of this extension, and its universal property, in $\bf Prof$?

It's been a long time since I tried to understand the deep meaning of the "Kan construction", or "nerve-realization" adjunction $$ \text{Lan}_y F \dashv N_F = \hom(F,1) $$ that exists among the left Kan extension of $F\colon \mathcal{A}\to \mathbf{D}$ ($\cal A$ small, $\bf D$ cocomplete) along the Yoneda embedding $y\colon {\cal A} \to \hat{\cal A}$. It can be expressed as $$ \text{Lan}_y F \dashv \text{Lan}_F y, $$ and this property seems pretty peculiar; especially if I think about the definition of a "Yoneda structure"[here, several comments in the discussion are mine].

1. Is there a reason why this is true?
2. What are other examples of a span of functors ${\bf C} \xleftarrow{G} {\cal A} \xrightarrow{F} {\bf B}$ such that $\text{Lan}_GF\dashv \text{Lan}_FG$?
3. My sensation is that this question acquires a (more?) meaning plunging the 2-category $\bf Cat$ in $\bf Prof$ in the usual way. The functor $N_F = \hom(F,1)$ is the image of $F$ via the canonical 2-functor $\varphi^{(-)} : {\bf Cat}^\text{co} \to \bf Prof$, and $N_F$ has a right adjoint $\hom(1,F)$; the Kan construction amounts to say that this extends to a triple of adjoints $$ \text{Lan}_yF = \varphi_F^! \dashv \varphi^F\dashv \varphi_F. $$ What's the meaning of this extension, and its universal property, in $\bf Prof$?

It is a long time that I try to understand the deep meaning "Kan construction", o "nerve-realization" adjunction $$ \text{Lan}_y F \dashv N_F = \hom(F,1) $$ that exists among the left Kan extension of $F\colon \mathcal{A}\to \mathbf{D}$ ($\cal A$ small, $\bf D$ cocomplete) along the Yoneda embedding $y\colon {\cal A} \to \hat{\cal A}$. It can be expressed as $$ \text{Lan}_y F \dashv \text{Lan}_F y, $$ and this property seems pretty peculiar; especially if I think about the definition of a "Yoneda structure"[here, several comments in the discussion are mine].

1. Is there a reason why this is true?
2. What are other examples of a span of functors ${\bf C} \xleftarrow{G} {\cal A} \xrightarrow{F} {\bf B}$ such that $\text{Lan}_GF\dashv \text{Lan}_FG$?
3. My sensation is that this question acquires a (more?) meaning plunging the 2-category $\bf Cat$ in $\bf Prof$ in the usual way. The functor $N_F = \hom(F,1)$ is the image of $F$ via the canonical 2-functor $\varphi^{(-)} : {\bf Cat}^\text{co} \to \bf Prof$, and $N_F$ has a right adjoint $\hom(1,F)$; the Kan construction amounts to say that this extends to a triple of adjoints $$ Lan_yF = \varphi_F^! \dashv \varphi^F\dashv \varphi_F $$ what's the meaning of this extension, and its universal property, in $\bf Prof$?

It's been a long time since I tried to understand the deep meaning "Kan construction", o "nerve-realization" adjunction $$ \text{Lan}_y F \dashv N_F = \hom(F,1) $$ that exists among the left Kan extension of $F\colon \mathcal{A}\to \mathbf{D}$ ($\cal A$ small, $\bf D$ cocomplete) along the Yoneda embedding $y\colon {\cal A} \to \hat{\cal A}$. It can be expressed as $$ \text{Lan}_y F \dashv \text{Lan}_F y, $$ and this property seems pretty peculiar; especially if I think about the definition of a "Yoneda structure"[here, several comments in the discussion are mine].

1. Is there a reason why this is true?
2. What are other examples of a span of functors ${\bf C} \xleftarrow{G} {\cal A} \xrightarrow{F} {\bf B}$ such that $\text{Lan}_GF\dashv \text{Lan}_FG$?
3. My sensation is that this question acquires a (more?) meaning plunging the 2-category $\bf Cat$ in $\bf Prof$ in the usual way. The functor $N_F = \hom(F,1)$ is the image of $F$ via the canonical 2-functor $\varphi^{(-)} : {\bf Cat}^\text{co} \to \bf Prof$, and $N_F$ has a right adjoint $\hom(1,F)$; the Kan construction amounts to say that this extends to a triple of adjoints $$ Lan_yF = \varphi_F^! \dashv \varphi^F\dashv \varphi_F $$ what's the meaning of this extension, and its universal property, in $\bf Prof$?