Let $F^\bullet : \mathcal{A} \to \mathcal{B}$ be a cohomological delta-functor which vanishes in degree strictly greater than $d$. Thus, $F^{d-\bullet}$ is a homological delta-functor.

Now assume that $F^n$ is effaceable (by injectives) for all $n \geq 1$, so that $F^\bullet$ is a universal delta-functor.

Then what can be said about $F^{d-\bullet}$ ? Is it coeffaceable (even if $\mathcal{A}$ does not necessarily have enough projectives) ? Is it universal ?