In Categories for Working Mathematician, Mac Lane describe a cartesian product as a limit for a functor F from a discrete category $|J|$ : Any cone from an object Z to F, is a collection of arrow from that object to some objects in the domain of F. He then consider a non discrete category J and derives an expression for the limit the domain is Set, i which case this amounts to a subset.
But we also have in general the injection $ d : |J| \rightarrow J$ from the discrete category $|J|$ derived from J where we strip everything except the identity.
Is it possible to recover the limit definition as a pushout along $d$ of the product ? I don't have the intuition as to why it would or would not be possible..
Conversely, I assume that the product can be recovered as pullback along $d$, as $|J|$ is equivalent to the comma category $*\downarrow (! :J \rightarrow 1) $ and pulling back preserves isos.
Can it also be recovered from the fact that limits are right kan extensions along $!$, and that right kan extension can themselves be defined by pointwise limit on the category of elements ?