I am wondering about an analog of the homotopy extension property in the setting of a pair $(\mathcal{C}, \mathcal{S})$ where $\iota : \mathcal{S} \to \mathcal{C}$ is a subcategory.


Explicitly, I say the pair $(\mathcal{C}, \mathcal{S})$ has the homotopy extension property if for any functors $F : \mathcal{C} \to \mathcal{D}$ and $G : \mathcal{S} \to \mathcal{D}$ along with a natural transformation $\eta : F \circ \iota \to G$ there exits a functor $G' : \mathcal{C} \to \mathcal{D}$ such that $G = G' \circ \iota$ and a natural transformation $\eta' : F \to G'$ restricting to $\eta$.


I am wondering if this notion has been studied and if there are any nice results on when such a pair has this homotopy extension property (HEP).


What happens if we replace "natural transformation" with "natural isomorphism"? I am actually interested in the version where $F, G : \mathcal{C} \to \mathcal{D}$ are specified along with a natural isomorphism over $\mathcal{S}$ which I want to extend to $\mathcal{C}$.


Some remarks:


(1) if $\iota$ is an equivalence then $(\mathcal{C}, \mathcal{S})$ has HEP


(2) let $I = \{ a \to b \}$ be the interval category. If $(\mathcal{S} \times I) \cup (\mathcal{C} \times \{ a \}) \to \mathcal{C} \times I$ has a left inverse then $(\mathcal{C}, \mathcal{S})$ has HEP where $\cup$ denotes the smallest subcategory containing both.



Homotopy extension property of subcategory
In the natural isomorphism case, such an inclusion functor is also called a cofibration of categories. In fact they are the cofibrations in the [canonical model structure](https://ncatlab.org/nlab/show/canonical+model+structure+on+Cat) on $Cat$, and they are characterized as the functors that are injective on objects (hence, in particular, include *all* actual subcategory inclusions).


I don't know what happens if you consider extending noninvertible transformations. In that case you would have two different notions of, say, "lax cofibration" and "colax cofibration".


