I would like to know if there is a special name for the following concept, papers that feature something similar or a general reference. Let $\mathcal{C}$ be a category and $\mathcal{D}$ a subcategory (or more generally a subclass of objects and morphisms). For a diagram $F:J\to\mathcal{C}$, say that the $\mathcal{D}$-limit of $F$ is a cone in $\mathcal{C}$ that satisfy a universal property respect to all cones whose vertex and morphisms to the vertex are in $\mathcal{D}$. $\mathcal{D}$-colimit would be the same concept but with cocones. It is clear that a $\mathcal{C}$-limit is precisely a usual limit, while any cone is a $\emptyset$-limit.


I am particularly interested in understanding in greater generality the behaviour of the examples in the following questions:


-[Do colimits of manifolds coincide with underlying colimits as topological spaces?](https://mathoverflow.net/questions/396340/do-colimits-of-manifolds-coincide-with-underlying-colimits-as-topological-spaces)


-[Are GIT's good categorical quotients just locally ringed space coequalizers?](https://mathoverflow.net/questions/59812/are-gits-good-categorical-quotients-just-locally-ringed-space-coequalizers)


-[Colimits of manifolds](https://mathoverflow.net/questions/38575/colimits-of-manifolds)


[M-complete category](https://ncatlab.org/nlab/show/M-complete+category) is certainly related but too specific.



Reference for certain categorical limits
This is such a natural notion, that I feel it must be written about somewhere, but I don't think it's in any of my usual sources like Borceux's books, Mac Lane's *Categories for the working mathematician*, or Emily Riehl's book *Category Theory in Context*.


It seems to me that such a notion, if previously invented, might be called a "relative limit" or "relative colimit." It feels like the sort of thing Grothendieck could have written about. Googling led me to writings of Lurie definining the analogous notion for $\infty$-categories, e.g., in *Higher Topos Theory* (4.3.1) and at [this link in Kerodon](https://kerodon.net/tag/02KF).


The theory of Mahavier limits is also related, as [this paper by Ittay Weiss spells out](https://pure.port.ac.uk/ws/files/8457717/MahavierLimitsUni.pdf).


