Let $F:\mathcal{C}\rightarrow\mathcal{D}$ be a full and faithful functor. Consider the corestriction $F:\mathcal{C}\rightarrow F(\mathcal{C})$ of $F$ to its image. Note that for $D\in F(\mathcal{C})$, we have $D=F(C)$ for some $C\in\mathcal{C}$: i.e. $D\cong F(C)$. This implies that the corestriction of $F$ to its image is an equivalence of categories.
Now, I don't understand the following:
Replacing $F(\mathcal{C})$ by the full subcategory $\mathcal{C}'\subset\mathcal{D}$ generated by all the objects $D\in\mathcal{D}$ isomorphic to an object of the form $F(C)$, one still has an equivalence $F:\mathcal{C}\rightarrow\mathcal{C}'.$
I don't follow this characterization of $\mathcal{C}'$: specifically, I am not sure what a "subcategory generated by all the objects etc." means. I know that $$\mathcal{C}'(D,D'):=\mathcal{D}(D,D')$$ for all $D,D'\in\text{Ob}(\mathcal{C}')$–but what are the object of $\mathcal{C}'$?
Can someone please give an explicit description of the full subcategory $\mathcal{C}'$?
Edit:
Is $\text{Ob}(\mathcal{C}')$ merely $\{D\in\text{Ob}(\mathcal{D})\ |\ (\exists C\in\mathcal{C})( D\cong F(C)\}$? Then, why say "generated"?
If $\mathcal{D}$ is a category then a subcategory $\mathcal{C}'$ of $\mathcal{D}$ is called full if for any two objects $X$ and $Y$ of $\mathcal{C}'$ we have $$ \mathcal{C}'(X, Y) = \mathcal{D}(X, Y) \,. $$ A full subcategory is uniquely determined by its class of objects. Given any class of objects $\mathcal{O} \subseteq \operatorname{Ob}(\mathcal{D})$ we can therefore talk about the full subcategory of $\mathcal{D}$ generated by $\mathcal{O}$. This is the unique subcategory $\mathcal{C}'$ of $\mathcal{D}$ with $\operatorname{Ob}(\mathcal{C'}) = \mathcal{O}$ and $\mathcal{C}'(X,Y) = \mathcal{D}(X,Y)$ for all objects $X, Y \in \mathcal{O}$.
In the given example we take $$ \mathcal{O} = \{ D \in \operatorname{Ob}(\mathcal{D}) \mid \text{there exists $C \in \operatorname{Ob}(\mathcal{C})$ with $D \cong F(C)$} \} \,. $$ The resulting full subcategory $\mathcal{C}'$ of $\mathcal{D}$ is known as the essential image of $F$.
Regarding your edit: I don’t think there is a particular reason why the verb “generated” is used here. One could also use a different verb which describes the situation. (I personally would talk about the full subcategory of $\mathcal{D}$ whose class of objects is given by $\mathcal{O}$.)
On a side note: The image $F(\mathcal{C})$ won’t necessarily be a subcategory of $\mathcal{D}$. So in general you won’t get an equivalence of categories between $\mathcal{C}$ and $F(\mathcal{C})$. (But if $F(\mathcal{C})$ is a subcategory of $\mathcal{D}$ then the corestriction of $F$ will indeed be such an equivalence of categories.)