Here's a simple example of a Reedy category $R$ for which the inclusion $R^+ \to R$ is not initial. Take $R$ to be the Reedy category freely generated by a single degree-decreasing map $1 \to 0$. The only degree-increasing maps are the identities, so $R^+ \to R$ is the object-set inclusion $\{0,1\} \to R$, which is not a weak homotopy equivalence, and hence not an initial functor.

(To be inclusive of both the 1-categorical and the ∞-categorical notions of initial functor, replace "a weak homotopy equivalence" by "injective on $\pi_0$".)

I don't know the actual definition, but what about 'elegant Reedy' categories?

Is your counterexample one of those?

@IanColey My example is not an elegant Reedy category. But its opposite is, and so is an example of an elegant Reedy category $C$ such that the inclusion $C^- \to C$ is not final.

Isn't the opposite of your category the same thing?

oh but the Reedy structure swaps around

That's right, so the opposite Reedy category has a single non-identity degree-increasing map.

Hang on, are you using initial to mean 'pullback along me doesn't change the colimit'*?

Okay your examples still work up to everyone's vocabulary

In my vocabulary "initial" relates to limits and "final" relates to colimits. But yes, the inclusions I gave fail to be either initial or final.

I've actually used the reedy structure on [1]^n in a paper haha

I was computing some kind of homotopy coend in a model category, and horror of horrors, it turns out that homotopy coends as such are not that well worked out in the literature. So George Raptis suggested this other technique, which worked quite well

@AlexanderCampbell Ok, thanks!