What is the explicit definition of this phrase in this theorem? From Mac Lane's Category Theory, what exactly is being assumed in the statement of this theorem?  What does the enclosed red box in the below picture mean?
I can see that $C$ has equalizers of all pairs of arrows means every pair of arrows in $C$ has an equalizer, but I don't understand what the next part is saying.  Can someone EXPLITICTLY explain what this means?


 A: We take products of objects in a category indexed by a set.  This is all he is saying:  if products always exist for index sets of certain sizes, you will have all colimits in the shape of $J$ (assuming coequalizer stuff).  And, of course, you might have all products, not just those of a certain size, but that is just gravy and does not violate any sufficient condition he gives.
Given an index set $I$, we may consider a set of objects of $\mathcal{C}$ indexed by $I$.  This would be a collection of objects $\{X_i : i \in I\}$, each $X_i$ an object of $\mathcal{C}$.  We may then speak of (or, ask for the existence of) the product $\prod_{i \in I} X_i$.  This is a "product indexed by $I$."  
Mac Lane is assuming that ALL such products exist whenever the index set is $\mathrm{obj}(J)$ or  $\mathrm{arr}(J)$.  This is really an assumption on the (roughly, modulo cardinalities) number of factors in such products.  If your object set is $\{a, b, c\}$ it certainly does not matter if you index a product of three terms as $X_a \times X_b \times X_c$ or $X_1 \times X_2 \times X_3$: it's still a product of three objects. 
