---
tag: NY3RGGYNNB
kind: statement
role: theorem
name: Adjoint functor theorem for posets
sources:
  - wikipedia: en:156411@r1364444923
    locator: Galois connection, section Connection to category theory > Adjoint functor theorem for posets
    note: adopted
    mention: wikipedia://en/156411#8339
    sections: [Statement]
inventory: [TGNT6XGD2V]
---

## Statement

Let $f : L \rightarrow P$ be a monotone function from a complete lattice to a poset. Then the following are equivalent:

1. f preserves all meets,
1. f has a left adjoint.

And similarly: f preserves all join if and only if f has an right adjoint.