Strict categorical localization is automatically a "2-localization"? This is a slightly pedantic question about the "2-categorical" nature of localization. Recall the definition:
Definition. A localization of a category $\cal C$ with respect to a class of morphisms $W$ is a category ${\cal C}[W^{-1}]$ together with a functor $q:{\cal C}\to {\cal C}[W^{-1}]$ such that


*

*$q$ sends the morphisms of $W$ to isomorphisms;

*for any functor $F:{\cal C}\to {\cal D}$ which sends the morphisms of $W$ to isomorphisms, there is a unique functor $G:{\cal C}[W^{-1}]\to {\cal D}$ such that $G\circ q=F$.


Note that I'm using the "strict" version of the definition, which characterizes the category ${\cal C}[W^{-1}]$, if it exists, up to a canonical isomorphism of categories (as opposed to a weaker version that only requires $G \circ q$ to be naturally isomorphic to $F$ and which only characterizes ${\cal C}[W^{-1}]$ up to equivalence).
The above very familiar definition can be rephrased as follows: condition (1) is equivalent to saying that for any category ${\cal D}$, the functor $-\circ q : {\cal D}^{{\cal C}[W^{-1}]} \to {\cal D}^{\cal C}$ factors through the full subcategory ${\cal D}^{({\cal C},W)}$ of ${\cal D}^{\cal C}$ consisting of those functors $F:{\cal C}\to {\cal D}$ which send the morphisms in $W$ to isomorphisms, and the universal property (2) states that the functor $-\circ q : {\cal D}^{{\cal C}[W^{-1}]} \to {\cal D}^{({\cal C},W)}$ is bijective on objects.
But what about natural transformations? It should follow from the above definition that localization is actually "2-categorical" in the sense that the functor $-\circ q : {\cal D}^{{\cal C}[W^{-1}]}\to {\cal D}^{\cal C}$ is fully faithful, so that $-\circ q : {\cal D}^{{\cal C}[W^{-1}]} \to {\cal D}^{({\cal C},W)}$ is an isomorphism of categories, not just a bijection between objects.
This would be immediate provided that $q$ is bijective on objects. It is probably obvious that the definition of localization forces this, but I can't see it. It is clear to me that a localization $q$ must be surjective on objects but I'm missing why it must be injective on objects. I'm sure there is a very simple reason, flying straight from the definition, that I'm missing.
Aside: one reason why someone might care about this detail comes from the theory of derivators. To show, for example, how derived categories give rise to derivators one needs to induce natural transformations between functors between derived categories.
 A: Let $\widetilde{\mathcal{C}}$ be the category with same objects as $\mathcal{C}$ and exactly one morphism between each pair of objects. Then there exists a unique functor $F:\mathcal{C} \to \widetilde{\mathcal{C}}$ that is the identity on objects. It sends all morphisms in $W$ to isomorphisms (since all morphisms of $\widetilde{\mathcal{C}}$ are isomorphisms). Thus it factors as $F = G q$ for a unique $G : \mathcal{C}[W^{-1}]\to\widetilde{\mathcal{C}}$. Since $F$ is the identity on objects, it follows that $q$ is injective on objects.
A: You can also see that the 2-dimensional universal property of q, that $-\circ q$ is fully faithful, follows directly from the 1-dimensional universal property you describe in your definition.  More generally any cocone in a 2-category (Cat here) which satisfies the 1-dimensional universal property of the colimit (coinverter here) also satisfies the 2-dimensional universal property so long as the 2-category you are working in has cotensors with 2, the category with two objects $0$ and $1$ and a single arrow $0 \to 1$.
Concretely this works in the above case as follows.  Let $q$ satisfies the 1-dimensional universal property and suppose we have a pair of functors $F,G:{\cal C}\to {\cal D}$ which both invert the W's and a natural transformation $\alpha:F \to G$.  Such a triple $(F,\alpha,G)$ uniquely correspond to a functor $\hat{\alpha}:C \to Ar(D)$ where Ar(D) is the arrow category of D, whose objects are arrows in D, and morphisms commuting squares (this is the cotensor of D with 2 in Cat).  Because isomorphisms in Ar(D), equally the functor category from 2 to D, are those nat. transformations which are pointwise isos it is clear the $\hat{\alpha}$ inverts the W's since both F and G do.  Thus you obtain a unique functor $\beta:{\cal C}[W^{-1}] \to D$ such that $\beta q = \hat{\alpha}$ and now postcomposing $\beta$ with the evident, and universal, two functor projections and natural transformation from $Ar(D)$ to $D$ gives the two functors and natural transformation from ${\cal C}[W^{-1}]$ to $D$ that you are after.
Apologies if the tex is madness as I can't see it on this computer.
A: The localisation of a category (ignoring size issues), when defined in the 'up to equivalence' way, is a coinverter in $Cat$. This is an example of a weighted colimit in the $Cat$-enriched category $Cat$. If one wants the 'up to isomorphism' version, then what you are dealing with is a strict coinverter. This obviously satisfies a slightly different universal property.
A: Considering the groupoids  $Grpd({C})$ this is maked by the  free category generate by the  morphisms of $\mathcal{C}$, their formal inverse and formal composition, and quotient this free category by the relations maked from the composition of $\mathcal{C}$. Then  considering the natural functor $F: \mathcal{C}\to Grp({C})$ that is the identity on objects, factorizing it follow  that your functor $q$ is a section on  object part is a section, then  injective.
Of course $q$ is surjective , this follow from the (strict) universal property (2), and considering the full category generated by the image of $q$.
