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, At this stage, we can already deduce uniqueness of the desired morphism, T is the initial algebra of ? ? + Id, as a (finitary) endofunctor on [Set, Set] f . By inspecting the colimit of the relevant initial chains, it can be shown that T (1) is the initial algebra of ? ? + Id +1 . Given the data, M has (? ? + Id +1)-algebra structure: the natural transformation Id ? M is merely the composite Id ? T ? M with the unit of T . By initiality, we have a unique (? ? + Id +1)-algebra morphism i : T (1) ? M as functors, such that i ? j = m

, the above adjunction, we get a (? ? + 1)-algebra morphism T (1) ? M , and thus in particular a T -module morphism T (1) ? M . We now apply this recipe to define unary multiterm substitution