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Let $C$ be a quasi-category. Then there is an imbedding $$C^{op} \times C \to \mathrm{Kan}$$ where $\mathrm{Kan}$ is the quasi-category of Kan complexes. This is essentially constructed in Lurie's book by choosing a model for $C$ as the nerve of a fibrant simplicial category $\mathfrak{C}$ and then taking the nerve of the ordinary Yoneda imbedding $\mathfrak{C} \times \mathfrak{C}^{op} \to \mathrm{Kan}$.

However, by the Grothendieck construction, the Yoneda imbedding should correspond to a left fibration (i.e., a quasi-category cofibered in Kan complexes) over $C \times C^{op}$. Is there a direct way to construct such a left fibration? I'm wondering if there is a way to do this without appealing to the theory of simplicial categories. I'd even be happy with something weakly equivalent to this left fibration in the covariant model structure.

In Lurie's book, a notion of bifibration (as Mike Shulman observes, this is elsewhere called a two-sided fibration) over a product $S \times T$ of simplicial sets is introduced, to correspond to the idea of a Kan complex functorial in two variables, but covariantly in one and contravariantly in another. I'm not very familiar with this theory, but a more general question would be how to turn a bifibration into a right or left fibration.

2 added 372 characters in body

Let $C$ be a quasi-category. Then there is an imbedding $$C \times  C^{op} \times C \to \mathrm{Kan}$$ where $\mathrm{Kan}$ is the quasi-category of Kan complexes. This is essentially constructed in Lurie's book by choosing a model for $C$ as the nerve of a fibrant simplicial category $\mathfrak{C}$ and then taking the nerve of the ordinary Yoneda imbedding $\mathfrak{C} \times \mathfrak{C}^{op} \to \mathrm{Kan}$.

However, by the Grothendieck construction, the Yoneda imbedding should correspond to a left fibration (i.e., a quasi-category cofibered in Kan complexes) over $C \times C^{op}$. Is there a direct way to construct such a left fibration? I'm wondering if there is a way to do this without appealing to the theory of simplicial categories. I'd even be happy with something weakly equivalent to this left fibration in the covariant model structure.

In Lurie's book, a notion of bifibration over a product $S \times T$ of simplicial sets is introduced, to correspond to the idea of a Kan complex functorial in two variables, but covariantly in one and contravariantly in another. I'm not very familiar with this theory, but a more general question would be how to turn a bifibration into a right or left fibration.

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# $(\infty, 1)$-Yoneda embedding via the Grothendieck construction

Let $C$ be a quasi-category. Then there is an imbedding $$C \times C^{op} \to \mathrm{Kan}$$ where $\mathrm{Kan}$ is the quasi-category of Kan complexes. This is essentially constructed in Lurie's book by choosing a model for $C$ as the nerve of a fibrant simplicial category $\mathfrak{C}$ and then taking the nerve of the ordinary Yoneda imbedding $\mathfrak{C} \times \mathfrak{C}^{op} \to \mathrm{Kan}$.

However, by the Grothendieck construction, the Yoneda imbedding should correspond to a left fibration (i.e., a quasi-category cofibered in Kan complexes) over $C \times C^{op}$. Is there a direct way to construct such a left fibration? I'm wondering if there is a way to do this without appealing to the theory of simplicial categories. I'd even be happy with something weakly equivalent to this left fibration in the covariant model structure.