Let $C$ and $D$ be categories and $F:C\rightarrow D$ a faithful functor which is full with respect to isomorphisms. This means that if $a,b\in C$ and $f:F(a)\rightarrow F(b)$ is an isomorphism in $D$, then $f$ is in the image of the functor $F$.
Suppose further that $D$ has all filtered colimits over diagrams in the image of $F$, so that $F$ induces a functor $\hat{F}:\text{ind-}C\rightarrow D$. Here $\text{ind-}C$ is the free ind-completion of $C$ obtained by formally adjoining all filtered colimits.
Question: Under what circumstances is $\hat{F}$ full with respect to isomorphisms?
Edit: Since posing the question, I've learned some more terminology from the nLab, and I can add some more restrictions. I am actually interested in the case when the image of $F$ in $D$ is closed under isomorphism, i.e. $F$ is an isofibration, equivalently $F(C)$ is a replete subcategory of $D$. Perhaps more usefully, we may assume that every object in $F(C)$ is a compact object in $D$. Finally, if it's relevant, we may assume that every arrow in $D$ is monic.
Motivation (from model theory): In generalized Fraïssé theory (a.k.a. Hrushovski constructions), one often works with a distinguished "strong substructure" relation $\preceq$ between finite $L$-structures in a specified class $K$. Some basic assumptions are that our class $K$ and the relation $\preceq$ are closed under isomorphism, and that $\preceq$ is reflexive and transitive. Then we have a notion of strong embedding ($f:A\rightarrow B$ is strong iff $f(A) \preceq B$), and the category $C_K$ whose objects are $K$ and whose arrows are the strong embeddings is a replete subcategory of the category $C_L$ of all $L$-structures and embeddings.
Now we are not just interested in the structures in $K$, but also the infinite structures which are built from them (the Fraïssé limit in particular), and we want to talk about when a finite structure $A$ in $K$ is a strong substructure of a structure $M$, which is a direct limit along a diagram of strong embeddings.
Clearly any structure appearing in the direct limit diagram should be a strong substructure of $M$, as should any strong substructure of such a structure, by transitivity. So we could take the strong embeddings into $M$ to be those which factor as a strong embedding into one of the structures in the diagram, followed by the inclusion into the direct limit. These are exactly the maps which appear in the construction of $\text{ind-}C_K$, so they're the ones which are in the image of $\hat{F}$ above.
But it's possible that the resulting notion is not isomorphism invariant. That is, there could be another direct limit $M'$ and an isomorphism (in $C_L$) $\sigma:M\rightarrow M'$ and a finite structure $A\preceq M$ such that $\sigma(A) \npreceq M'$.
A map from the formal colimit of a directed system $(A_i)$ to the formal limit of a directed system $(B_i)$ in $\text{ind-}C_K$ is a choice for each $A_i$ of a strong embedding into some $B_j$ such that all resulting diagrams commute. So given $\sigma:M\cong M'$ where $M$ and $M'$ are direct limits, $M = \text{colim}(A_i)$, $M' = \text{colim}(B_i)$, along strong embeddings, $\sigma$ is in the image of $\hat{F}$ if and only if each $A_i$ embeds strongly in some $B_j$ by a restriction of $\sigma$ and each $B_i$ embeds strongly in some $A_j$ by a restriction of $\sigma^{-1}$. This happens if and only if whenever $A\preceq M$, $\sigma(A) \preceq M'$. So the resulting strong substructure notion is isomorphism-invariant if and only if $\hat{F}$ is full with respect to isomorphisms.
Another way of putting it is that whenever we have two presentations of a structure $M$ as direct limits along strong embeddings, these two presentations "fit together".
What I'm looking for is a weak condition, or better a characterization, in terms of properties of the $\preceq$ relation, of when this happens. For example, it certainly happens if $A \preceq C$ and $A \subseteq B \subseteq C$ implies $A \preceq B$, which is often the case in interesting examples (for example, whenever $A \preceq -$ is definable by a universal theory with parameters from $A$).