  The main purpose of this paper is to show that the converse of the known implication weakly action representable implies action accessible is false. In particular we show that both action accessibility, as well as the (at least formally stronger) condition requiring the existence of all normalizers do not imply weakly-action-representability even for varieties. In addition we show that in contrast to both action accessibility and the condition requiring the existence of all normalizers, weakly-action representability is not necessarily inherited by Birkoff subcategories. 
A note on the relationship between action accessible and weakly action   representable categories