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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 $F$. enter image description here 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.

enter image description here 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?

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