Given a category $\mathcal{C}$ and two objects $A,B \in \mathcal{C}$, we can construct the "polynomial category" $\mathcal{C}[x : A \to B]$ constructed by adjoining an indeterminate arrow $x : A \to B$.
Now, suppose instead we have another category $\mathcal{C}'$ and a faithful functor $F : \mathcal{C} \hookrightarrow \mathcal{C}'$ and a morphism $f \in \mathcal{C}'(A,B)$. Clearly (glossing over the details here, of course, as in the previous example), we should be able to construct the category $\mathcal{C}(f)$ given by adjoining the arrow $f$ from $\mathcal{C}'$.
The former example of a polynomial category shows up in many different places in the literature -- and what I have proposed for the category $\mathcal{C}(f)$ seems like a natural generalization of the idea. Has this or otherwise a similar construction shown up in the literature at all? If so, what is some of the terminology that has been used for such a construction?
Such a construction has proven useful for generating interesting examples in my work, but I haven't seen anyone doing anything analogous in the literature. This of course generalizes the idea of an extension of a field by an element.