I am interested in collecting in this question a list of properties a functor $F$ may have and what those properties imply for left and right adjoints, $F^L$ and $F^R$, assuming they exist. There are different types of functors and different types of categories, but let us begin in complete generality. If this format seems to work, I'd be interested in specializing to abelian and derived categories, and also including higher categories, in separate lists - we'll see (of course, if someone is eager to add and populate the other lists, feel free!).

The motivation here is that, as a "working mathematician", I find it annoying to have to search the literature everytime something like this comes up. In homage to Maclane, I therefore dub this post "Categories for the Lazy Mathematician".

The format of the list is this: give a property of $F:C\to D$, and what it implies for $F^L:D \to C$ or $F^R: D \to C$, as the case may be. Let us try to be concise in each entry of the list. Give any helpful details about each claim as a separate answer, e.g. if you feel a definition is obscure, you can provide it, or if you want to prove an implication or give a reference, do so there. Naturally, there are many more properties of functors than the ones I came up with below. Please add them if they are interesting, even if you do not know the implications.

**General categories**

- $F$ faithful
- $\Leftrightarrow$ the unit $\mathrm{id_C} \to F^R \circ F$ is a pointwise monomorphism
- $\Leftrightarrow$ the counit $F^L\circ F\to \mathrm{id_C}$ is a pointwise epimorphism

- $F$ full
- $\Leftrightarrow$ the unit $\mathrm{id_C}\to F^R\circ F$ co-splits pointwise in $C$
- $\Leftrightarrow$ the counit $F^L\circ F\to \mathrm{id_C}$ splits pointwise in $C$

- $F$ is fully faithful
- $\Leftrightarrow$ the counit $F^L \circ F \to \mathrm{id_C}$ is an isomorphism
- $\Leftrightarrow$ the unit $\mathrm{id_C} \to F^R \circ F$ is an isomorphism

- (Assuming $C,D$ are essentially small) $F$ essentially surjective $\Rightarrow$ the induced functor on presheaf categories $$\Delta_F\colon[D^{\text{op}},\mathbf{Set}]\to [C^{\text{op}},\mathbf{Set}]$$ is faithful and conservative, so its unit $\mathrm{id}\to\Pi_F\Delta_F$ is pointwise mono and its counit $\Sigma_F\Delta_F\to\mathrm{id}$ is pointwise epi, as above. In this case, $\Delta_F$ is both monadic and comonadic.
- $F$ is a wide inclusion, i.e. $F$ is faithful and essentially surjective $\Rightarrow$
- $F$ dominant $\Rightarrow$ same conclusion as $F$ essentially surjective (since $F$ is essentially surjective on Cauchy completions and the Cauchy completion has the same presheaf category)
- (Assuming $C$ has finite limits) $F$ conservative $\Leftrightarrow$ the counit is a strong epimorphism
- $F$ separable $\Rightarrow$
- $F$ injective on objects $\Rightarrow$
- $F$ preserves limits $\Rightarrow$ $F^L$ exists (assuming $C$ is complete and "small enough")
- $F$ preserves colimits $\Rightarrow$ $F^R$ exists (assuming $C$ is cocomplete and "small enough")
- $F$ preserves $\kappa$-directed colimits $\Rightarrow$ $F^L$ preserves $\kappa$-presentable objects

**Abelian categories**

$F$ exact $\Rightarrow$ $F^R$ preserves injectives

$F^L$ (resp. $F^R$) exists $\Rightarrow$ $F$ and $F^L$ (resp. $F^R$) are additive

**Monoidal categories**

- $F$ lax monoidal $\Rightarrow$ $F^L$ colax monoidal (doctrinal adjunction, holds in much greater generality). Dually, $F$ colax monoidal $\Rightarrow$ $F^R$ lax monoidal.

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