Formal criteria for adjoint functors

Freyd's adjoint functor theorem^[1] — Let ${\displaystyle G:{\mathcal {B}}\to {\mathcal {A}}}$ be a functor between categories such that ${\displaystyle {\mathcal {B}}}$ is complete. Then the following are equivalent (for simplicity ignoring the set-theoretic issues):

1. G has a left adjoint.
2. ${\displaystyle G}$ preserves all limits and for each object x in ${\displaystyle {\mathcal {A}}}$, there exist a set I and an I-indexed family of morphisms ${\displaystyle f_{i}:x\to Gy_{i}}$ such that each morphism ${\displaystyle x\to Gy}$ is of the form ${\displaystyle G(y_{i}\to y)\circ f_{i}}$ for some morphism ${\displaystyle y_{i}\to y}$.

Kan criterion for the existence of a left adjoint — Let ${\displaystyle G:{\mathcal {B}}\to {\mathcal {A}}}$ be a functor between categories. Then the following are equivalent.

1. G has a left adjoint.
2. G preserves limits and, for each object x in ${\displaystyle {\mathcal {A}}}$, the limit ${\displaystyle \lim({(x\downarrow G)\to {\mathcal {B}}})}$ exists in ${\displaystyle {\mathcal {B}}}$.^[2]
3. The right Kan extension ${\displaystyle G_{!}1_{\mathcal {B}}}$ of the identity functor ${\displaystyle 1_{\mathcal {B}}}$ along G exists and is preserved by G.

Moreover, when this is the case then a left adjoint of G can be computed using the left Kan extension.^[2]