MHB Is the Left-Adjoint Functor Preserving Colimits of Functors?

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Here's this week's problem!

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Problem. Let $L : \mathcal{C} \to \mathcal{C}$ be a left-adjoint functor from category $\mathcal{C}$ to category $\mathcal{C'}$. Show that if $F : \mathcal{D} \to \mathcal{C}$ is a functor such that $\operatorname{colim} F$ is an object of $\mathcal{C}$, then $L(\operatorname{colim} F)$ is a colimit of $L \circ F : \mathcal{D} \to \mathcal{C'}$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

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No one answered this week's problem. You can find my solution below.

Spoiler

If $\mathcal{C}\overset{L}{\underset{R}{\rightleftarrows}}\mathcal{C'}$ is an adjunction, then for every object $Y$ in $\mathcal{C'}$,

\begin{align}
\operatorname{Hom}_{\mathcal{C}}(L(\operatorname{colim} F, Y) &\approx \operatorname{Hom}_{\mathcal{C'}}(\operatorname{colim} F, RY)\\
&\approx \operatorname{colim} \operatorname{Hom}_{\mathcal{C'}}(F, RY)\\
&\approx \operatorname{colim} \operatorname{Hom}_{\mathcal{C}}(L\circ F, Y)\\
&\approx \operatorname{Hom}_{\mathcal{C}}(\operatorname{colim}(L\circ F), Y).
\end{align}

Hence by Yoneda's lemma,

$$L(\operatorname{colim} F) \approx \operatorname{colim}(L\circ F).$$