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In Wikipedia's article on Kan extensions 1, in the view of Kan extensions as colimits, I am confused about the notation: $(Lan_F X)(b) = \varinjlim_{f:Fa \to b} X(a)$.

Wikipedia says that the colimit is over the category $F \downarrow b$, but the functor $X$ is from $\mathbf{A}$ to $\mathbf{C}$. I am a bit confused about this, I understand that the functor "inside" the colimit notation should be defined over the category where we are taking it. Is there any place where I can read about this limit/colimit notation?

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Well, one extends $X$ to a functor $F \downarrow b \to C$ by composing with the forgetful functor $F \downarrow b \to A$. This is also suggested by the notation $\varinjlim_{Fa \to b} X(a)$, one forgets about the morphism $Fa \to b$ and only remembers $a$.

For example, if $f : X \to Y$ is a continuous map, $F$ is a presheaf on $Y$, then $f^{-1} F$ is the presheaf on $X$ defined by $V \mapsto \varinjlim_{U \subseteq f^{-1}(V)} F(V)$, i.e. it is the left Kan extension of $F$ along $f^{-1} : \mathrm{Open}(Y)^{op} \to \mathrm{Open}(X)^{op}$.

For basics about category theory, I don't recommend Wikipedia, but rather the books by Mac Lane, Kashiwara + Shapira, Mac Lane + Moerdjik, Borceaux.

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