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I'm wondering if anyone knows a reference for the following construction: let $\k$ be a field, say, and assume for convenience everything below is $\k$-linear, and that every category is essentially small. Hereafter "right exact" means "which commutes with finite colimits".


Let $\A$ be a finitely cocomplete category, and $\B$ be an arbitrary (still $\k$-linear and essentially small) category. Suppose we are given a functor
$$
\iota :\A \longrightarrow \B$$
which I'm happy to assume essentially surjective if that helps. Define the relative $\A$-cocompletion of $\B$ to be a category $\B\_\A$ equipped with a functor $\nu:\B\rightarrow \B\_\A$, universal for the following properties:


1. $\B\_\A$ is finitely cocomplete
2. for any finitely cocomplete $\C$, restriction along $\nu$ induces a natural equivalence between: (a) right exact functors $\B\_\A \rightarrow \C$ and (b) just functors $\B \rightarrow \C $ such that the composition $\A \rightarrow \B \rightarrow \C$ is right exact.


In words, I want to complete $\B$ under finite colimit, without duplicating those already existing in the image of $\A$. If $\A$ is $Vect$ this should just be the free finite cocompletion.


Remember that the finite cocompletion of $\B$ is the full-subcategory of (linear) presheaves on $\B$ which are finite colimits of representables. I believe $\B\_\A$ is then the full subcategory of that, of those presheaves having the property that their restriction along $\iota$ is representable. An example which is an inspiration for this definition is the construction of the category of modules over a monad, from the category of free modules, see e.g. <https://ncatlab.org/nlab/show/Eilenberg-Moore+category#AsColimitCompletionOfKleisliCategory>.


I'm in the process of checking that myself, and it's probably just formal, but I would much rather have a reference where it's done properly.



Relative cocompletion of a category
This is a special case of the general construction of cocompletions that preserve existing colimits. The general statement can be found as Theorem 6.23 of Kelly's *Basic Concepts of Enriched Category Theory*, and more explicitly as Proposition 11.4 and Theorem 11.5 of Fiore's *Enrichment and Representation Theorems for Categories of Domains and Continuous Functions* (in the case of small cocompletions, though the result is easily modified to work with a class of colimits instead). In summary, for classes $\Phi, \Psi$ of colimits for which the small category $\mathbf B$ is $\Phi$-cocomplete, there is a conservative $\Psi$-cocompletion $\widehat {\mathbf B}\_\Phi$ of $\mathbf B$ preserving the $\Phi$-colimits. This means that the restriction of the (restricted) Yoneda embedding $\mathbf B \to \widehat {\mathbf B}\_\Phi$ is $\Phi$-cocontinuous and exhibits a bijection between $\Phi$-cocontinuous functors $\mathbf B \to \mathbf C$ into cocomplete categories $\mathbf C$, and $\Phi$- and $\Psi$-cocontinuous functors $\widehat {\mathbf B}\_\Phi \to \mathbf C$. Explicitly, $\widehat {\mathbf B}\_\Phi$ is the subcategory of the category of presheaves on $\mathbf B$ which are $\Psi$-colimits of representables taking $\Phi$-cocones to limiting $\Phi$-cones.


In your setting, take $\Phi$ to be the class of colimits in the image of $\iota : \mathbf A \to \mathbf B$, and take $\Psi$ to be the class of finite colimits. Then $\widehat {\mathbf B}\_\Phi$ is exactly the finite cocompletion of $\mathbf B$ relative to $\iota$.


