# Iterated endofunctors

Suppose $F : \mathbb{C} \rightarrow \mathbb{C}$, with the following constraints:

• $F^{n+1}(\mathbb{C})$ is a subcategory of $F^{n}(\mathbb{C})$
• for all objects $X \in \mathbb{C}$ there exists a monomorphism $i : F^nX \rightarrow F^{n+1}X$
• morphisms $f \in \mathbb{C}(X, Y)$ are mapped in such a way, that they commute with previous $i$ morphisms (that is, $(i_X)_X$ are natural transformation $\mathbb{id}\Rightarrow F$)

where $F^n$ denotes $n$ repeated applications of $F$. Let $F^\infty(X)$ be an object with a mono $X \rightarrow F^\infty(X)$, which is universal in the sense, that for any other object $Y$ and mono $X\rightarrow Y$ there exists a unique mono $Y \rightarrow F^\infty(X)$.

Does anyone know how this construction is commonly called in category theory? I tried googling for a bit, but couldn't find anything.

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See ncatlab.org/nlab/show/transfinite+construction+of+free+algebras and the references there. But I think they almost never have the mono condition. –  Martin Brandenburg Apr 10 '13 at 9:29