# Kan 延拓的两个例子　`ex.kan-extension`
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> English: **draft** · machine-drafted, pending author review; the Chinese page is authoritative.

layer 12 · proposition · cluster 伴随与反射 (`cat.adjoint`) · Category Theory

## Abstract
Two examples showing that Kan extensions absorb the earlier notions. (1) A diagram $D : I \to \mathcal{C}$ has a colimit iff the left Kan extension of $D$ along $p : I \to \mathbf{1}$ exists, and then $(p_!D)(\ast) = \varinjlim_{i \in I} D(i)$. (2) $F : \mathcal{C} \to \mathcal{D}$ has a right adjoint iff the left Kan extension of $1_{\mathcal{C}}$ along $F$ exists, and then $G = F_!1_{\mathcal{C}}$, the unit being the transpose of $1_{\mathcal{C}} \implies G \circ F$.

> 续见 nodes/ex.kan-extension.en.2.md