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


1. $q$ sends the morphisms of $W$ to isomorphisms;
2. 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.



Strict categorical localization is automatically a "2-localization"?
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.


