The following concept seems to be useful:



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> **Definition.** Let $\mathbf{J}$ and $\mathbf{C}$ denote categories, and suppose we're given a functor $F:\mathbf{J} \rightarrow \mathrm{End}(\mathbf{C}).$ A *generalized cocone* from $F$ is an object $X$ of $\mathbf{C}$ together with, for each object $Y$ of $\mathbf{J}$, a morphism $\varphi\_Y :F(Y)(X) \rightarrow X$, such that for each $f : Y \rightarrow Y'$ in $\mathbf{J}$, we have: $$\varphi\_Y=\varphi\_{Y'} \circ F(f)(X).$$ Given cocones $(X,\varphi)$ and $(X',\varphi')$, a *morphism* $g : (X,\varphi) \rightarrow (X',\varphi')$ is just a morphism $g : X \rightarrow X'$ such that for all $Y \in \mathbf{J}$, we have $g \circ \varphi\_Y = \varphi'\_Y\circ F(Y)(g)$.
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> A *generalized colimit* is an initial generalized cocone.
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The main examples are:


1. Given a functor $F:\mathbf{J} \rightarrow \mathbf{C}$, we can get a functor $G:\mathbf{J} \rightarrow \mathrm{End}(\mathbf{C})$ by assigning to every object $Y$ of $\mathbf{J}$ the constant functor whose value is $F(Y)$. A cocone from $F$ is the same thing as a generalized cocone from $G$.
2. Let $\mathbf{J} =\{Y\_0,Y\_1\}$ denote the discrete category with two objects. Let $F : \mathbf{J} \rightarrow \mathrm{End}(\mathbf{Set})$ denote a functor such that $F(Y\_0)$ is a constant endofunctor with value $S \in \mathbf{Set}$. Then a generalized colimit of $F$ is just an $F(Y\_1)$ algebra freely generated by $S$. For example, we can define $\mathbb{N}$ this way, by taking $S = \mathbb{1}$ and $F(Y\_1) = \mathrm{id}\_\mathbf{Set}$.



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> **Question.** What are generalized cocones/colimits really called, and where can I learn more about them?
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What are generalized cocones/colimits really called?
I don't know references. But here is a reformulation of this notion which might be useful.


I will assume that $\mathbf{C}$ is small-cocomplete, $\mathbf{J}$ is small and that $F : \mathbf{J} \times \mathbf{C} \to \mathbf{C}$ is a functor (this is equivalent to a functor $\mathbf{J} \to \mathrm{End}(\mathbf{C})$). Then, for every $X \in \mathbf{C}$ we have the functor $F(-,X) : \mathbf{J} \to \mathbf{C}$ and may take its colimit $G(X) := \mathrm{colim}\_{\mathbf{J}} F(-,X)$. Since colimits are functorial, we get a functor $G : \mathbf{C} \to \mathbf{C}$. The observation is:


* A generalized cocone from $F$ is the same as a [$G$-algebra](https://ncatlab.org/nlab/show/algebra+for+an+endofunctor), i.e. an object $X \in \mathbf{C}$ equipped with a morphism $G(X) \to X$
* This is actually an isomorphism of categories. In particular, generalized colimits of $F$ are precisely the [initial $G$-algebras](https://ncatlab.org/nlab/show/initial+algebra+of+an+endofunctor).


In particular, we may apply the theory of initial algebras in order to understand generalized colimits. For instance, [Adámek’s theorem](https://ncatlab.org/nlab/show/initial+algebra+of+an+endofunctor#AdameksTheorem) shows that a sufficient condition for the generalized colimit of $F$ to exist is that $G$ preserves colimits of $\omega$-chains. And for this it is sufficient that for all objects $I \in \mathbf{J}$ the functor $F(I,-) : \mathbf{C} \to \mathbf{C}$ preserves colimits of $\omega$-chains.


