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Suppose I have a category $\cal C$ and a surjective function $f:\operatorname{obj}({\cal C})\to X:\bf Set$. Is there a natural way to construct a category $f(\cal C)$ with $\operatorname{obj}(f({\cal C}))=X$ reflecting the category structure of $\cal C$? I am aware of the concept of an image of a category under a functor $F:\cal C\to D$, but this presupposes a category structure on the codomain, which I don't have.

Since I'm not sure what options are available here, I can't say what properties to expect or require, except that if $f$ is a bijection then $f(\cal C)$ should be isomorphic to $\cal C$ (and $f$ will be the object part of the isomorphism functor).

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    Yes, there's a universal such thing. Basically the morphisms are strings of morphisms in $C$ which would be composable if the objects of $C$ were identified the way $f$ identifies them, subject to the minimal relations compatible with the relations between morphisms in $C$. – 2017-01-07
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    @QiaochuYuan That sounds right. It seems very similar to the image of a functor in that it is composing things that were not previously composable, but I'm not sure what category is being used in the background (the free category on $X$?). Could you put that in an answer? How do you state the universal property that this satisfies? – 2017-01-07

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Consider the object functor $\mathcal Cat\overset{\mathrm{ob}}\to\mathcal Set$ from the category of small categories to the category of sets.

An $\mathrm{ob}$-sink structure on a set $X$ is collection of set-functions $\mathrm{ob}(\mathcal C_i)\xrightarrow{f_i}X$ for small categories $\mathcal C_i$. An $\mathrm{ob}$-lift of this sink structure is a collection of functors $\mathcal C_i\xrightarrow{F_i}\mathcal C$ whose images under $\mathcal Cat\overset{\mathrm{ob}}\to\mathcal Set$ factor as $\mathrm{ob}(\mathcal C_i)\xrightarrow{f_i} X\xrightarrow{f}\mathrm{ob}(\mathcal C)$. A morphism of $\mathrm{ob}$-lifts is simply a functor $\mathcal C\xrightarrow G\mathcal D$ so that $\mathcal C_i\xrightarrow{G_i}\mathcal D$ factor as $\mathcal C\xrightarrow{F_i}\mathcal C\xrightarrow{G}\mathcal D$ with $X\xrightarrow{g}\mathrm{ob}(\mathcal D)$ factoring as $X\xrightarrow{f}\mathrm{ob}(\mathcal C)\xrightarrow{\mathrm{ob}(G)}\mathrm{ob}(\mathcal D)$.

A semi-final $\mathrm{ob}$-lift is an initial object in the category of $\mathrm{ob}$-lifts; it is called final if the morphism $X\xrightarrow{f}\mathrm{ob}(\mathcal C)$ is an isomorphism.

In particular, every small(!) $\mathrm{ob}$-sink structure on a set $X$ has an final $\mathrm{ob}$-lift by declaring $\mathrm{Hom}(A,B)$ to be the set of finite sequences $(\alpha_j)$ of morphisms in the union of morphisms $\bigcup_i\mathrm{ar}(\mathcal C_i)$ that are compatible in the sense that if $\alpha_j\in\mathcal C_i$ and $\alpha_{j+1}\in\mathcal C_k$, then $f_i(\mathrm{dom}(\alpha_j))=f_k(\mathrm{cod}(\alpha_k))$, modulo the induced composition relation on paths coming from the compositions in each category $\mathcal C_i$.

The free category on $X$ is the particular case of the final lift of the empty $\mathrm{ob}$-sink on $X$.

For more general background, see the articles on final lifts and topological categories at the nlab; note the category of small categories is not made topological by the object functor because the latter is not a faithful functor - this is reflected in the restriction that only small $\mathrm{ob}$-sinks have final lifts.