An opposite limit theorem

This is something that those who ought to know these things usually do know, and eventually figure out in any case. So if it isn't usually part of your work to know these things, don't fret; I really ought to have learned it much sooner!

Suppose $D$ is a category with a terminal object, say $z$, and let $F:D\to C$ be any functor. Then the natural transformation $F0 : F\to Fz$ is initial among the category of objects $x$ of $C$ with natural transformations $F\to x$.

That is all.