# Limit as a pushout

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 $$\mathbf{Set}$$, in 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.

So I have three questions :

• 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?