# special adjoint functor theorem　`thm.saft`
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> English: **draft** · machine-drafted, pending author review; the Chinese page is authoritative.

layer 13 · theorem · cluster 伴随与反射 (`cat.adjoint`) · Category Theory

## Abstract
Let $\mathcal{D}$ be small and complete. A functor $G : \mathcal{D} \to \mathcal{C}$ has a left adjoint iff it preserves all limits; when it does, one may take $F(X) = \varprojlim_{(Y,f) \in (X \downarrow G)} Y$. So preservation of limits, which is automatic for right adjoints, is also sufficient under these hypotheses. The construction reads: $F(X)$ is the object of $\mathcal{D}$ closest to $X$, and every $X \to G(Y)$ factors through $X \to G(FX) \to G(Y)$.

> 续见 nodes/thm.saft.en.2.md