| Theorem: Let $\mathcal{C}$, $\mathcal{D}$, $\mathcal{E}$ be categories and $U : \mathcal{D}\to \mathcal{C}$, $V : \mathcal{E}\to \mathcal{C}$, $F : \mathcal{D} \to \mathcal{E}$ be functors such that $V \circ F = U$. Suppose $U, V$ have left adjoints and $\mathcal D$ admits coequalizers. If $V$ reflects split epimorphisms to regular epimorphisms, then $F$ has a left adjoint. | |