I would like concrete descriptions of the adjoint functors that arise in a geometric morphism induced by a functor between the base categories of categories of presheaves.
Suppose we have functor $F$ from a category $\mathbb{C}$ to a category $\mathbb{D}.$ Let $\mathbb{D}^{op}$ denote the opposite category to $\mathbb{D}.$ As usual we write $\textbf{Set}^{\mathbb{D}^{op}}$ to denote the category of presheaves on $\mathbb{D}.$ In this case, if I understand correctly, there will be a funtor $F^{*}$ from $\textbf{Set}^{\mathbb{D}^{op}}$ to $\textbf{Set}^{\mathbb{C}^{op}}$ which sends an object $\mathbb{D}^{op}\overset{X}{\rightarrow}\textbf{Set}$ of $\textbf{Set}^{\mathbb{D}^{op}}$ to the precomposition $\mathbb{C}^{op}\overset{X\circ F^{op}}{\longrightarrow}\textbf{Set}.$ Also $F^{*}$ sends an arrow $X\overset{\alpha}{\rightarrow}Y$ of $\textbf{Set}^{\mathbb{D}^{op}}$ to the natural transformation $(X\circ F^{op})\overset{\beta}{\rightarrow}(Y\circ F^{op})$ with $\beta_{c}=\alpha_{F(c)}$ for each object $c$ of category $\mathbb{C}.$
If I understand correctly from here then $F^{*}$ is part of an essential geometric morphism, and so there is a right adjoint $F_{*}$ of $F^{*}$ and there is a left adjoint $F_{!}$ of $F^{*}.$
My question is, how exactly do the functors $F_{*}$ and $F_{!}$ work on objects AND ON ARROWS ? Ideally I would like the answer to minimize the usage high level concepts, because I am quite inexperienced with adjoint functors and I do not know about Kan extensions.