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The fact is that $F^{*}$ always has both a left and a right adjoint. These are called left and right Kan extensions along $F$. The same is true if you replace $\mathbf{Set}$ by any category with small limits and colimits.

The fact is that $F^{*}$ always has both a left and a right adjoint. These are called left and right Kan extension along $F$. The same is true if you replace $\mathbf{Set}$ by any category with small limits and colimits.

${}$2. Kan extensions. Let $F: \mathbf{A} \to \mathbf{B}$ be any functor between small categories. Then there's an induced functor $$ F^{*}: {[\mathbf{B}, \mathbf{Set}]} \to {[\mathbf{A}, \mathbf{Set}]} $$ defined by composition with $F$. Thus, ${F^{*}(Y) = Y\circ F}$ for every $Y \in {[\mathbf{B}, \mathbf{Set}]}$.

${}$2. Kan extensions. Let $F: \mathbf{A} \to \mathbf{B}$ be any functor between small categories. Then there's an induced functor $$ F^{*}: {[\mathbf{B}, \mathbf{Set}]} \to {[\mathbf{A}, \mathbf{Set}]} $$ defined by composition with $F$. (Here ${[\mathbf{B}, \mathbf{Set}]}$ means the category of functors from $\mathbf{B}$ to $\mathbf{Set}$, sometimes denoted ${\mathbf{Set}}^{\mathbf{B}}$.)

$\mbox{}$1. Forgetful functors between categories of algebras. Any time you have a category $\mathcal{A}$ of algebras, such as Group, Ring, Vect, ..., the forgetful functor $\mathcal{A} \to \mathbf{Set}$ has a left adjoint. What's not quite so well-known is that you don't have to forget all the structure; that is, the codomain doesn't have to be Set.

$\mbox{}$2. Kan extensions. Let $F: \mathbf{A} \to \mathbf{B}$ be any functor between small categories. Then there's an induced functor $$ F^{*}: [\mathbf{B}, \mathbf{Set}] \to [\mathbf{A}, \mathbf{Set}] $$ defined by composition with $F$. Thus, $F^{*}(Y) = Y\circ F$ for every $Y\in [\mathbf{B}, \mathbf{Set}]$. (Here $[\mathbf{B}, \mathbf{Set}]$ means the category of functors from $\mathbf{B}$ to $\mathbf{Set}$, sometimes denoted ${\mathbf{Set}}^{\mathbf{B}}$.)

${}$1. Forgetful functors between categories of algebras. Any time you have a category $\mathcal{A}$ of algebras, such as Group, Ring, Vect, ..., the forgetful functor $\mathcal{A} \to \mathbf{Set}$ has a left adjoint. What's not quite so well-known is that you don't have to forget all the structure; that is, the codomain doesn't have to be Set.

${}$2. Kan extensions. Let $F: \mathbf{A} \to \mathbf{B}$ be any functor between small categories. Then there's an induced functor $$ F^{*}: {[\mathbf{B}, \mathbf{Set}]} \to {[\mathbf{A}, \mathbf{Set}]} $$ defined by composition with $F$. Thus, ${F^{*}(Y) = Y\circ F}$ for every $Y \in {[\mathbf{B}, \mathbf{Set}]}$.