[Algebras for endofunctors](https://ncatlab.org/nlab/show/algebra+for+an+endofunctor) bridge the gap between functors acting *on* a category and structures defined *in* it. An algebra for an endofunctor $F$ is instantiated by some morphism $Fa \to a$, and more crucially a *morphism* of algebras is a map $a\to b$ between their codomains making the evident diagram commute. This construction is intrinsically homotopical, so it makes good sense to loosen the restriction from just endofunctors.


Do you know of any writing about algebras for more general [transfors](https://ncatlab.org/nlab/show/transfor)? Seems that many 2-categorical results about algebras over monads, lax morphisms, et cetera, would enjoy a more full account of transfors and their algebras.



Algebras for general transfors
The category of algebras for an endofunctor $F:C\to C$ is the lax limit of the 2-functor $[F]:D\to Cat$, where $D$ is the 2-category freely generated by one object and one endomorphism. Accordingly, it seems natural to me that more general lax limits could be regarded as categories of "algebras for transfors".


For instance, if instead we take $D$ to be the 2-category freely generated by one object, two endomorphisms $f,g$ of that object, and a 2-cell $f\to g$, then a diagram $D\to Cat$ consists of two endofunctors $F,G$ of a category $C$ and a transformation $\alpha:F\to G$. Then an object of the lax limit could be regarded as an "algebra for $\alpha$". Concretely, this consists of an object $X$ with maps $FX\to X$ and $GX\to X$ making a triangle commute with $\alpha\_X$. Of course this is equivalent to just making $X$ a $G$-algebra, so algebras for a natural transformation aren't a new thing.


However, other lax limits give examples of other important and related structures. For instance, the category of monad-algebras for a monad, and the category of pointed-algebras for a pointed endofunctor, are both lax limits. And one can likewise write down different kinds of lax limits of higher categories to represent other algebra-like structures.


