Understanding Colimits: A Proof and Explanation

[Jim Kata]

I have been trying to teach myself some category theory, and have been working through some proofs. I didn't understand the proofs I read about proving colimits (in the case of a small categories) can be given in terms of coproducts and coequalizers. Here is my attempt at a proof. I would appreciate someone to correct my mistakes and explain the aspects I don't understand. I'm sorry if it is a bit disconbobulating since I'm not sure how to draw diagrams in latex. Let [tex] \mathcal{F} : \mathcal{B} \rightarrow \mathcal{A} [/tex] be a diagram where [tex] \mathcal{B}[/tex] is a small category. Let [tex]X_j[/tex] be an object in [tex] \mathcal{B}[/tex] (we can index it i guess because [tex] \mathcal{B}[/tex] is a small category?)

let [tex] \varphi : X_{j} \rightarrow X_{l} [/tex]

so [tex]\mathcal{F}(\varphi) : \mathcal{F}(X_{j}) \rightarrow \mathcal{F}(X_{l})[/tex]

Since the coproduct exists for every [tex]X_j[/tex]

there exists [tex]i_j:X_j \rightarrow \coprod_{Obj \mathcal{B}}B[/tex]

so there are two morphisms [tex]\mathcal{F}(i_j):\mathcal{F}(X_j) \rightarrow \mathcal{F}(\coprod_{Obj \mathcal{B}}B)[/tex]
and [tex]\mathcal{F}(i_l\varphi):\mathcal{F}(X_j) \rightarrow \mathcal{F}(X_l)\rightarrow \mathcal{F}(\coprod_{Obj \mathcal{B}}B)[/tex]

Using the existence of the coequalizer we have the cocone [tex](\phi,Q)[/tex]

where [tex]\phi(X_j)= q\circ\mathcal{F}(i_j): \mathcal{F}(X_j) \rightarrow Q[/tex] and by the universal property of the coequalizer we get the universal property of the cocones. I guess my problem is I don't see how I ever used the universal property of the coproduct and I'm not sure I used the small category part right?

 

[Jhero]

I'm guessing it is because the coproduct is indexed by the objects of a small category but I don't understand how that works here. Thanks for any help!