Recursion categories of coalgebras

We construct recursion categories from categories of coalgebras. Let $F$ be a nontrivial endofunctor on the category of sets that weakly preserves pullbacks and such that the category $\textbf{Set}_F$ of $F$-coalgebras is complete. The category $\textbf{Set}_F$ may be embedded in the category $\mathbf{Pfn}_F$ of $F$-coalgebras and partial morphisms, which is a $P$-category that is prodominical but not dominical in general. An existence theorem of A. Heller is applied to certain subcategories of $\textbf{Pfn}_F$ to obtain examples of recursion categories of coalgebras.