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.^[3][4][5]

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