---
tag: CKTDR0VE8D
kind: concept
name: Solution set condition
msc: [18A40]
sources:
  - url: https://en.wikipedia.org/wiki/Adjoint_functors
---

## Summary

The solution set condition on a [[functor|28NQ0KRHD3]] $G \colon \mathcal{D} \to \mathcal{C}$ asks the following.
For each object $c$, a mere set of morphisms $c \to G d$ is enough to factor all of them.
It is a smallness condition.
Together with completeness and preservation of limits, it gives the existence of a left [[adjoint|TZHF6ZM92S]] in Freyd's adjoint functor theorem.

## Explanations

### L1 human

To build a free group on a set $X$, one would like to look at all maps from $X$ into all groups at once.
There are too many groups for that to be a set.
But every such map lands in the subgroup generated by its image, and that subgroup is not much bigger than $X$.
So a set of small groups is enough to see all maps from $X$.
The solution set condition asks for such a set in general.