Consider the category **Cat** as a *concrete category* over **Set** $\times$ **Set** via the functor   

*U* : **Cat** $\rightarrow$ **Set** $\times$ **Set**, defined by


*U*$(\mathbf A \xrightarrow{F} \mathbf B) = ($*Ob*$(\mathbf A)\xrightarrow{F\_O}$ *Ob*$(\mathbf B)$ , *Mor*$(\mathbf A)\xrightarrow{F\_M}$ *Mor*$(\mathbf B))$, 


where $F\_O$ is the restriction of $F$ to objects and $F\_M$ is its restrictions to morphisms.


**Cat** is the category of all *small* categories in the sense that their *objects* and *morphisms* form sets (not classes).


It is clear that this *concrete category* is *transportable*. **But is this category uniquely transportable?**


A concrete category $(\mathbf A, U)$ over $\mathbf X$ is said to be **(uniquely) transportable** provided that for every $\mathbf A-$object A and every $\mathbf X-$isomorphism $UA\xrightarrow{k}X$ there exists a (unique) $\mathbf A-$object B with $UB=X$ such that $A\xrightarrow{k}B$ is an $\mathbf A-$isomorphism.



$\mathbf C\mathbf a\mathbf t$ as a concrete category
Let $X = (O, M)$ be a pair of sets, and let $A$ be a category with set of objects $O\_A$ and set of morphisms $M\_A$. Suppose given a pair $k = (k\_0, k\_1)$ of isomorphisms $k\_0: O\_A \to O$, $k\_1: M\_A \to M$. Let $s\_A, t\_A: M\_A \to O\_A$ be the source and target functions for $A$; then the only way to define the source and target functions $s, t: M \to O$ on the lifted structure over $X$ in such a way that $k$ preserves source and target data is by setting 


$$s = k\_0 \circ s\_A \circ k\_1^{-1}, \qquad t = k\_0 \circ t\_A \circ k\_1^{-1}$$ 


(that's what I meant by "conjugating" in my comment). 


It's the same idea to define the lifted structure $i, c$ for identity and composition. If $i\_A, c\_A$ are the identity and composition on $A$ and $k$ is to preserve identity and composition data, then there's only one way to define them: put 


$$i = k\_1 \circ i\_A \circ k\_0^{-1}, \qquad c = [(f, g) \mapsto (k\_1 \circ c\_A)(k\_1^{-1} f, k\_1^{-1} g)]$$ 


where $f, g$ is any composable pair in the lifted source-target structure over $X$. 


Of course unique transport of structure via conjugation is a simple idea which applies much more generally, as hinted in Peter's second comment. 


