nLab
solution set condition

Idea

The solution set condition appears as part of the hypothesis in Freyd’s General Adjoint Functor Theorem.

Statement

A functor $F:C\to D$ satisfies the solution set condition if for every object $Y$ of $D$ there exists a small set $I$ and an $I$-indexed family of morphisms $\{f_i:Y\to F(X_i){\}}_{i\in I}$ such that any morphism $h:Y\to F(X)$ can be factored as

$$F(t)\circ f_i:Y\stackrel{f_i}{\to }F(X_i)\stackrel{F(t)}{\to }F(X)$$F(t) \circ f_i : Y \stackrel{f_i}{\to} F(X_i) \stackrel{F(t)}{\to} F(X)

for some $t:X_i\to X$ and some $i$.

This is a smallness condition in that the family is required to be indexed by a small set.