Nil_2(Grp) is weakly action representable with an abelian weak representing object but neither action representable nor locally algebraically cartesian closed.
Van der Linden,Non-associative algebras, New Perspectives in Algebra, Topology and Categories (M
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Weak action representability of 2-nilpotent groups
Nil_2(Grp) is weakly action representable with an abelian weak representing object but neither action representable nor locally algebraically cartesian closed.