This post is meant to get some possible reference on a topic that I do not see very much in literature.
Well, in "The Joy of Cats" the category of $T$-spaces is introduced. Given a covariant functor $T$ from a category $\mathfrak{C}$ to the category of sets, the authors say that a $T$-space is a pair $(X,K)$ where $X$ is an object of $\mathfrak{C}$ and $K \subseteq T(X)$. Regarding the morphisms, a function $f: X \to Y$ goes to a function $T(f): T(X) \to T(Y)$ in such a way to send elements of $K$ to elements of $K'$. This choice implies the covariance of the functor $T$, but nothing prevents to use a contravariant functor in the above definition, and to change suitably the notations (see http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf, Def. 5.40, page 73).
Now, it seems obvious to generalize the perspective. Let $\mathfrak{D}$ be a category and $T: \mathfrak{C} \to \mathfrak{D}$ a covariant functor. Then I call a $T$-space a pair $(X,K)$ where $X$ is an object of $\mathfrak{C}$ and $K$ is a subobject of $T(X)$ in the category $\mathfrak{D}$. Is there a natural choice for morphisms in such a case? If yes, have these categories been already studied elsewhere? Finally, what happens when choosing contravariant functors, namely is it possible to define in a natural way morphisms in this case?
Rephrasing Def. 5.40, pg. 73: Let $\boldsymbol{X}$ and $\boldsymbol{Set}$ be categories (the latter is the usual suspect). Then take $$T:\boldsymbol{X}\to \boldsymbol{Set}$$ to be a [covariant] functor. Then the category of $T$-spaces is just: $$\color{red}{\boldsymbol{Spa(T)}} := \Bigg\{\bigg\{\color{red}{(X,\alpha)}\text{ }\bigg|\text{ }\alpha\subseteq T(X)\bigg\},\text{ }\text{ }\bigg\{\color{red}{(X,\alpha)\to (Y,\beta)}\text{ }\bigg|\text{ }\big(T(X)\to T(Y)\big)(\alpha)\subseteq \beta\bigg\}\Bigg\}$$ $$= \bigg\{Obj(\boldsymbol{Spa(T)}),\text{ } Mor(\boldsymbol{Spa(T)})\bigg\},$$ where of course $X,Y\in \boldsymbol{X}$ are free.
By the defining condition for the T-maps (i.e. morphisms in $\boldsymbol{Spa(T)}$), it should be clear that $T$ must be covariant, since otherwise $T(f)$ would have reversed domain and codomain and hence wouldn't be able to accept $\alpha\subseteq T(X)$. One could swap $\alpha$ and $\beta$ in the condition to force contra-variant to work.
Replace $\alpha \leftrightarrow K$, $\beta \leftrightarrow K'$, and $\boldsymbol{\mathfrak{C}}\leftrightarrow \boldsymbol{X}$ to get your notation.
To generalize, let's repeat the one liner with red texts with appropriate changes:
For covariant functor $T:\boldsymbol{X}\to\boldsymbol{Y}$, define:
$\color{red}{\widetilde{\boldsymbol{Spa(T)}}} := \Bigg\{\bigg\{\color{red}{(X,\alpha)}\text{ }\bigg|\text{ }\alpha:=[Y\hookrightarrow T(X)]_{\sim}\bigg\},$ $$\bigg\{\color{red}{(X_1,\alpha_1)\to (X_2,\alpha_2)}\text{ }\bigg|\text{ }\exists \iota_{\alpha_1\alpha_2}:\big[\big(T(X_1)\to T(X_2)\big)\circ \iota_{\alpha_1}(Y_1)\hookrightarrow T(X_2)\big]_{\sim}\hookrightarrow \alpha_2\bigg\}\Bigg\},$$ where we have equivalence classes of monomorphisms $[\cdot \hookrightarrow \cdot\cdot]_{\sim}$ defining a subobject of $\cdot \cdot$. In the second definition we need to define a subobject of a subobject, but $T(f)$ doesn't operate on subobjects and may need to be monic itself for the composition to be monic. This may impose another restriction. At this point, a picture helps:
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It remains to show this definition is independent of representative, $\iota_{\alpha_1}:Y_1\hookrightarrow T(X_1)$, but this can be done. Existence of such an $\iota_{\alpha_1'\alpha_2}$ implies existence of $\iota_{\alpha_1\alpha_2}$ [Exercise].
This concludes my attempt at generalization. I have not seen this considered in any other texts (also have not seen $T$-spaces considered either). The closest thing I've seen are representations of functors with universal elements etc., but these are elements $u\in T(X)$ not subsets, so there is a distinction.