Adjoint functor ($L⊣R$)

An adjunction is a natural bijection $\mathcal{D}(LB,A)\cong \mathcal{C}(B,RA)$.

For $F⊣G$, the unit $\eta :1\to GF$ and counit $\epsilon :FG\to 1$ are the natural transformations corresponding to identity morphisms under the adjunction. They satisfy ${\epsilon }_{F}F\eta =1_F$ and $G\epsilon {\eta }_G=1_G$.

For $F⊣G$, the right adjoint $G$ is full and faithful exactly when the counit $FG\to 1$ is an isomorphism. Dually, $F$ is full and faithful exactly when the unit $1\to GF$ is an isomorphism.

The right Kan extension along $K:\mathcal{C}\to \mathcal{D}$ is the right adjoint to precomposition by $K$ between suitable functor categories.

The colimit form of the Special adjoint functor theorem says that a colimit-preserving functor from a locally small, cocomplete, well-copowered category with a small generating family to a locally small category has a right adjoint.

A category with finite products is cartesian closed when every product functor $-\times A$ has a right adjoint $(-{)}^A$, called exponentiation by $A$.

An object $A$ of a cartesian closed category is tiny when exponentiation $(-{)}^A$ itself has a right adjoint.