Producing a new category by identifying isomorphic objects In category theory it is common to talk about objects up to isomorphism. Two objects $X$ and $Y$ are isomorphic if and only if there exist a pair of arrows $f : Y \leftarrow X$ and $g : X \leftarrow Y $ so that $f \circ g $ is equal to $\mathrm{id}_Y$ and $g \circ f $ is equal to $\mathrm{id}_X$.
Suppose we have a category $\mathcal{C}$, I'm wondering whether it makes sense to define "$\mathcal{C}/{\simeq}$" ... the idea being that we simply identify all the isomorphic objects by fiat. I'm pretty sure that composition of morphisms means that we can combine and flatten the various homsets related to objects in the same equivalence class in a well-defined way.
I'm interested in performing this operation so I can talk about object sameness in $\mathcal{C}/{\simeq}$ rather than isomorphism in $\mathcal{C}$.
I tried hunting around for what this notion might be called, but the obvious name quotient category refers to something different. The notion of a quotient category seems to "thin out" individual homsets, but leave the objects of the overall category intact.
 A: Such a quotient could certainly be defined. The problem is that it would not do what you seem to think that it would. Given two isomorphic object, there is usually way more than one isomorphism between them. So if you do what you suggest, you end up identifying every automorphism of a given objects with the identity, the equivalence relation on morphisms becomes very complicated (something like : $f$ is equivalent to $g$ if and only if there are morphisms $h_1, ..., h_n$ and isomorphisms $u_0, u_1, ..., u_n$ such that $g = h_1 \cdots h_n$ and $f = u_0 h_1 u_1 h_2 u_2 \cdots h_n u_n$), and the resulting category does not look like $\mathcal C$ at all in general. Also, you need to be careful with set-theoretic questions. Namely, the resulting category does not stay in the same universe as $\mathcal C$ : morphism sets are guaranteed to stay in $\mathcal U$ only if $\mathcal C$ is $\mathcal U$-small (as is usually the case with this kind of construction, like localizations of categories). If $\mathcal C$ has Hom-sets in $\mathcal U$, then the Hom-sets of the quotient will in general be in a larger universe (namely, in $\mathcal U'$ such that $\mathcal U \in \mathcal U'$).
For all these reasons, one does not usually do such a construction. As @paul blart math cop pointed out, one usually takes a skeleton of the category, by choosing one object in each isomorphism class, and looking at the full subcategory $\mathcal S$ on these objects. Then, each time you consider an object of $\mathcal C$, you can choose a fixed isomorphism with the representative of its equivalent class in the skeleton. You can see that the inclusion of $\mathcal S$ into $\mathcal C$ is an equivalence of categories, so working with skeletons is basically the same as working "up to equivalence of categories". Note that there is a set-theoretic caveat (if you want to see it as a caveat) in this context too : although a skeleton has Hom-sets in the same universe as $\mathcal C$, the choice of a skeleton (if there is no canonical choice for your given category) requires the use of the axiom of choice for the isomorphism classes of $\mathcal C$, which are in general in a larger universe than the Hom-sets of $\mathcal C$. And the choices of isomorphisms with elements of the skeleton, if you want to do these all at once, would require the same kind of thing.
