Adjoint situation induced on presheaves I just read that every functor $\phi : C \to D$ induces the well-known adjoint situation
$\phi_! \dashv  \phi^\ast : \hat{D} \to \hat{C}$.
Well, this is not so well-known to me, so could someone please describe this adjunction in more detail or point to a reference?
 A: It is actually more general than that.
Take $\varphi \colon \mathcal C \to \mathcal D$ a functor between (small) categories, and let $\mathcal E$ be any category. The functor $\varphi$ induces a functor $\varphi^\ast \colon \mathcal E^\mathcal D \to \mathcal E^\mathcal C$ defined as $- \circ \varphi$, i.e. $\varphi^\ast (F) = F\circ \varphi$.
Then, if $\mathcal E$ is cocomplete (meaning it admits all small colimits), then $\varphi^\ast$ admits a left adjoint $\varphi_!$, called the left Kan extension functor. We even have a formula for this left adjoint:
$$\varphi_! (F) \colon d \mapsto \operatorname{colim}\left( (\varphi \downarrow d) \to \mathcal C \overset F \to \mathcal E \right).$$
Dually, if $\mathcal E$ is complete (i.e. has all small limits), then $\varphi^\ast$ admits a right adjoint $\varphi_\ast$, called the right Kan extension functor, for which we have the following formula:
$$\varphi_\ast (F) \colon d \mapsto \operatorname{lim}\left( (d\downarrow \varphi) \to \mathcal C \overset F \to \mathcal E \right).$$
In your case, $\mathcal E = \mathsf{Set}$ is complete and cocomplete, so both adjunctions occur.
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