Let $F:C\to D$ be a “nice” functor (for example, $H_*(-;\mathbb{Z}):\mathbf{Top}\to \mathbf{Ab}^{\mathbb{Z}}$). Now assume that we have a category $O$ internal to $C$. Is there a canonical way to construct a category $F_*O$ which is internal to $D$?

In the case where $C$ and $D$ are monoidal and $O$ is not internal, but only enriched, the crucial property $F$ should fulfill is monoidality, as we need morphisms $Fx\otimes Fy\to F(x\otimes y)$ when defining the composition. Here, it seems that there must be a transformation of pullbacks $Fx\times_{Fz} Fy\to F(x\times_z y)$, but maybe this is unnecessarily complicated.

I would be happy if one can construct such a pushforward for the above example $H_*(-;\mathbb{Z})$. In my case, I had to replace a $\mathbf{Top}$-enriched category $A$ by a $\mathbf{Top}$-internal category $B$ which is “homotopy equivalent” to the original one (when considering $\mathrm{ob}(A)$ as a discrete space), and I want to say something like “$H_*A$ and $H_*B$ are isomorphic as $\mathbf{Ab}^\mathbb{Z}$-enriched categories”.