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title: Functoriality of Kan extensions

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# Functoriality of Kan extensions


## Weighted colimits

$V$ a complete and cocomplete symmetric monoidal category

Recall that, given $\phi:J^{op}\to V$ and $D:J\to C$,  the weighted colimit is defined, if it exists, as the (unique up to iso) solution of 
$$C(colim_\phi D,B) \cong [C^{op},V](\phi,C(D-,B))$$

## Left Kan extensions

$K:A\to C$ and $G:A\to B$

Left Kan-extensions are defined as 
$$(Lan_KG)c = \int_a C(Ka,c)\bullet Ga = colim_{C(K-,c)}G$$

Spelling this out the definition of a weighted colimit, this means that  $Lan_KG$ is the (up to iso) unique solution of the equation
$$B((Lan_KG)c,b) \cong [A^{op},V](C(K,c),B(G,b))$$

**Remark:** The equation above is quite intuitive: Natural transformations between the cone under $c$ and the cone under $b$ are classified by $Lan_KG$. This is really the same idea as always when it comes to colimits.

With $b=Lan_KG$, we obtain the counit

$$C(K-,c)\to B(G,(Lan_KG)c)$$

## Functoriality of Left Kan extensions

$f:c\to c'$ induces $(Lan_KG)f : (Lan_KG)c\to (Lan_KG)c'$ defined as follows.

First, we know, see [(Kelly,(1.29))](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf), that $C(Ka,-)$ is a functor. That is, there is an arrow

$$C(Ka,-): C(c,c') \to [C(Ka,c),C(Ka,c')]$$

This, in turn, induces an arrow 

$$(Lan_KG)c \to (Lan_KG)c'$$

via (abbreviating $L=Lan_KG$)

\begin{align}
C(c,c')
& \rightarrow [A^{op},V](C(K,c),C(K,c')) & C(K,-) \textrm{ is a functor} \\
& \to [A^{op},V](C(K,c),B(G,Lc')) & \textrm{counit} \\
& \cong B(Lc,Lc') & \textrm{Def of colimit}
\end{align}

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## Various notes

$Lan_KG$ is, if it exists, the unique L such that
$$C(Ka,c) \to B((Lan_KG)Ka,(Lan_KG)c) \to B(Ga,(Lan_KG)c)$$

is an iso, natural in $a$, for all $c\in C$.
-->

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References: Chapter 4.1 of [Kelly](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf).