There is a notion of equivalence of categories which is the functor $F:\mathcal{C} \to \mathcal{D}$ such that there is a functor $G:\mathcal{D} \to \mathcal{C}$ such that $FG \cong id\_{\mathcal{D}}$ and $GF \cong id\_{\mathcal{C}}$ where $\cong$ means natural isomorphism (natural transformation with each component being the isomorphism). One can also check equivalence of categories without referring to $G$: it suffices that $F$ is *full*, *fauthul* (faithful means injective on arrows between fixed objects, full means surjective on arrows between $F(C\_1)$ and $F(C\_2)$) and *essentially surjective* (surjective on objects, up to isomorphism). Note that from being full and faithul follows that $F$ is *essentially injective* on objects (meaning that if $F(C\_1)$ and $F(C\_2)$ are isomorphic in $\mathcal{D}$ then $C\_1$ and $C\_2$ are isomorphic in $\mathcal{C}$). Is there an example of two categories $\mathcal{C}$ and $\mathcal{D}$ with the properties:  

-there is a functor $F:\mathcal{C} \to \mathcal{D}$ being essential bijection on objects (meaning essentially surjective and essentially injective in the above sense)  

-for each two objects $C\_1,C\_2 \in \mathcal{C}$ there is an *abstract* bijection between $Hom\_{\mathcal{C}}(C\_1,C\_2) \simeq Hom\_{\mathcal{D}}(F(C\_1),F(C\_2))$ meaning that this bijection is not implemented by $F$.
To be more precise: for each pair of objects $C\_1,C\_2 \in \mathcal{C}$ we can find a map $T\_{C\_1,C\_2}:Hom\_{\mathcal{C}}(C\_1,C\_2) \to Hom\_{\mathcal{D}}(F(C\_1),F(C\_2))$ being a bijection of sets and this map is not of the form $F(f)$ for $f \in Hom\_{\mathcal{C}}(C\_1,C\_2)$?



Equivalence of categeories-variants of definition
Let $D$ be the category whose only object is $\mathbb{N}$ and the morphisms are all functions $\mathbb{N} \to \mathbb{N}$. Let $C$ be the subcategory whose morphisms are bijections $\mathbb{N} \to \mathbb{N}$.


The inclusion $F : C \to D$ has the properties you ask for.


