§ 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.