Need an example of a category whose products are indexed by these sets.

Question

From a Theorem that states: For categories $J$ and $C$, if $C$ has equalizers of all pairs of arrows and "all products indexed by the sets obj($J$) and arr($J$)", then $C$ has a limit for every functor $F:J \rightarrow C.$

Can someone give a simple example of a category $J$ and $C$ such that "all products indexed by the sets obj($J$) and arr($J$)"?

I'm trying to understand what "all products indexed by ..." means and I'm not making progress.

Answer 1

You are misreading the sentence. The phrase "all products indexed by the sets obj($J$) and arr($J$)" is a noun phrase, not a verb phrase. In other words, the statement is

$C$ has ... all products [that are] indexed by the sets obj($J$) and arr($J$).

That is, every collection of objects of $C$ indexed by $\mathrm{obj}(J)$ or $\mathrm{arr}(J)$ has a product.