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?**

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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.}

Algebraic Theories: A Categorical Introduction to General Algebra. Cambridge University Press 2011], Example 13.17.2 gives thecategory $\Sigma$-Algof all '$\Sigma$-algebras' (perhaps not needless to say for some readers, these are not the '$\sigma$-algbras' of measure theory) asan example of a uniquely-transportable concrete category. $\endgroup$3more comments