Composition of left kan extensions

Question

Let $i: P\hookrightarrow Q$ and $j: Q\hookrightarrow R$ be inclusions of posets, and $M: P\to C$ where $C$ is a co-complete category. Does anyone have a reference for the fact that ${\rm Lan}_{j\circ i}(M) = {\rm Lan}_j({\rm Lan_i}(M))$? Or a slick proof?

Answer 1

First prove the following.

Lemma 1. For any functors $f : \mathbf C \to \mathbf D$ and $g : \mathbf D \to \mathbf E$ with respective left adjoints $L_f : \mathbf D \to \mathbf C$ and $L_g : \mathbf E \to \mathbf D$, the composite $L_fL_g$ is a left adjoint for $gf$.

Hint: it is a one-line proof.

Fact 2. If $\mathcal E$ is a cocomplete category and $p : A \to B$ a functor between small categories, left Kan extensions form a functor $\operatorname{Lan}_p : [A,\mathcal E] \to [B,\mathcal E]$, left adjoint to the restriction $p^\ast : [B,\mathcal E] \to [A,\mathcal E]$.

This is actually my definition of left Kan extensions, but if you have seen the notion with initial 2-arrows, you just have to translate all that in terms of adjunctions.

Lemma 3. For $p: A \to B$ and $q : B \to C$, one has $\operatorname{Lan}_{qp} \simeq \operatorname{Lan}_q \operatorname{Lan}_p$.

Just apply lemma 1 to $(qp)^\ast = p^\ast q^\ast$. Please remark that it gives a isomorphism of functors, not an equality (which actually could make no sense, depending on your framework and definition of left Kan extensions.)

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P.S. : I change notations compared to your question, just to emphasize that this has nothing to do with posets or embeddings.