§ Limit/Colimit/Cone/Cocone: the arrows are consistent!
I sometimes wonder if a product is a limit or a colimit (not really, because
I remember that limits are product + equalizers, but it makes for a nice story
nonetheless). I realised that the arrows of a cone/co-cone are always consistent.
Since a cone has arrows out of the apex, the universal cone is given by arrows into the apex
to be able to compose arrows
Similarly, since a co-cone has arrows into the apex, the universal co-cone must have arrows
out of the apex into the apex of another co-cone. Thus a terminal object must be a limit/cone,
since we want arrows into it (by universality), and that is compatible if the cone itself has arrows
out of the apex, ie, a limit, since a limit is a product and thus has projections out of it.