On Sheaves in Geometry and Logic MacLane and Moerdijk show (Theorem VII.2.2 pag. 359) that a functor $F:\mathbb{C}\rightarrow\mathbb{D}$ between small categories extends to an essential geometric morphism between the categories of presheaves. As their notation felt quite cryptic I tried to prove it in a slightly different way, but I'm not really sure about it (I guess otherwise I wouldn't be asking for a check)
The preliminary step (Corollary I.5.2, pag. 43) is to show that the Yoneda embedding allows to extend any functor $A:\mathbb{C}\rightarrow\mathcal{E}$, where $\mathbb{C}$ is small and $\mathcal{E}$ is cocomplete, to a functor $L_A:\mathbf{Set}^{\mathbb{C}^{op}}\rightarrow\mathcal{E}$ which happens to be a left adjoint for $R_A$ such that for $E\in\mathcal{E}$ one has $R_A(E)=\operatorname{Hom}_\mathcal{E} (A(-),E)$. $L_A$ is also unique up to natural isomorphism.
Now consider $F:\mathbb{C}\rightarrow\mathbb{D}$ and denote by $F^*$ the functor $-\circ F^{op}:\mathbf{Set}^{\mathbb{D}^{op}}\rightarrow\mathbf{Set}^{\mathbb{C}^{op}}$:
By the uniqueness (up to iso) of the extension it follows immediately $F^*\simeq L_{F^*y}\dashv R_{F^*y}$.
On the other hand, by Yoneda lemma \begin{equation} R_{yF}(Q)=\operatorname{Nat}(y(F(-)),Q)\simeq F^*(Q) \end{equation} naturally in $Q$ and hence $L_{yF}\dashv F^*\dashv R_{F^*y}$, which is the geometric morphism extending $F$.
You have to show that the adjunction $F_!\dashv F^*\dashv F_*$ exists: given $F : C \to D$ there is a functor $$F^* : [D,{\bf Set}]\to [C, {\bf Set}]$$ (the "inverse image", this name is self-explanatory), acting as precomposition and sending a presheaf $P : D \to \bf Set$ into the presheaf $PF :C \to \bf Set$. This functor now has a left adjoint and a right adjoint, characterized as left and right Kan extensions along $F$ (Borceux tome I and Mac Lane's CWM have excellent introductions to this notion).
I would like to point out that Not only theSE Kan extensions exist (because $\bf Set$ is cocomplete) but they are pointwise, as you can write the co/limits (or the coends <3) that define them. See §2.1 of my coend note for further infos and do not hesitate to ask more details!