What are generalized cocones/colimits really called? The following concept seems to be useful:

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 $g \circ \varphi_Y = \varphi'_Y\circ F(Y)(g)$.
A generalized colimit is an initial generalized cocone.

The main examples are:


*

*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 $G$.

*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 $F(Y_1) = \mathrm{id}_\mathbf{Set}$.

Question. What are generalized cocones/colimits really called, and where can I learn more about them?

 A: 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, 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.


In particular, we may apply the theory of initial algebras in order to understand generalized colimits. For instance, Adámek’s theorem 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.
