Kan extensions specify the adjoint structures between $\mathbf{Sets^{C^{op}}}$ and $\mathbf{Sets^{D^{op}}}$, where there exists a functor $f:\mathbf{C} \to \mathbf{D}$ and $\mathbf{C}$ and $\mathbf{D}$ are small categories. $(\infty,1)$-Kan extension is studied in higher topos theory, model categories, and homotopy theory such as homotopy Kan extensions.


A special case is the locally cartesian closed category. The triple adjunctions can be $\sum\_f \dashv f^{\*} \dashv \prod\_f$. 


My questions:


1.Are the above left and right adjoint of $f^{\*}$ unique ? Or there are other such adjoint compositions for $f^{\*}$ ?


2.There are [examples](http://ncatlab.org/nlab/show/Kan+extension#Examples) in nlab. But are there other cases other than $\mathbf{Sets^{I}}$ and $\mathbf{Sets/I}$ known with such structures ? 


3.What are higher analogies of special cases for the adjointness relations ?



Kan extensions and special cases
Adjoints are unique, and therefore, the functors $\Sigma\_f$ and $\Pi\_f$ are unique. As for examples of Kan extensions, take a look at the nlab page <http://ncatlab.org/nlab/show/examples+of+Kan+extensions>. I don't understand your second and third questions, but here's an example. Let $\mathbf{C}$ and $\mathbf{D}$ be ordinary categories with a functor $f:\mathbf{D}\to\mathbf{D}^\prime$. There is a functor $\mathbf{C}\to\mathbf{C}^\mathbf{D}$, and composition with $f$ gives a functor $f^\*:\mathbf{C}^{\mathbf{D}^\prime}\to\mathbf{C^D}$. A left adjoint to $f^\*$ is called *the left Kan extension along $f$*. If $\mathbf{D^\prime}$ is the one-object category, then the theory of left Kan extensions reduces to the theory of colimits (see Section 4.3 of HTT), with an analagous understanding of right Kan extensions.


