Let $\mathcal{C}$ and $\mathcal{D}$ be two equivalent categories. Furthermore, assume $\mathcal{C}$ is enriched over a monoidal category $(\mathcal{M}, \otimes)$. Can one use the equivalence to enrich $\mathcal{D}$ over $(\mathcal{M}, \otimes)$?

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    $\begingroup$ You have in mind some forgetful functor from $M$-enriched categories to ordinary categories. What is it? Are you applying $\text{Hom}(1, -)$ to each hom object? $\endgroup$ – Qiaochu Yuan Oct 9 '15 at 23:44
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    $\begingroup$ If the chosen lax monoidal "forgetful" functor $\mathcal{M} \to \mathbf{Set}$ is an isofibration then you can transport enrichments along equivalences. If you are happy to neglect isomorphisms then you don't even need the isofibration assumption – but that is somehow less interesting. $\endgroup$ – Zhen Lin Oct 13 '15 at 22:19

I think most category theorists would answer "yes, obviously", and not bother to write down a proof. But presumably that isn't sufficiently convincing, since you ask the question, so let me try to make it a bit more explicit with some big words. (-:

An $M$-enriched category with set of objects $A$ is equivalently a lax functor $A_{ch}\to B M$, where $A_{ch}$ is the chaotic category with $A$ as objects (exactly one morphism between any two objects) and $B M$ is the 1-object bicategory associated to $M$. A function $f:A\to B$ induces a functor $f_{ch}: A_{ch} \to B_{ch}$, hence by precomposition a map $f^*$ from $M$-categories with object set $B$ to $M$-categories with object set $A$, such that $f^*\underline{B}(a_1,a_2) = \underline{B}(f(a_1),f(a_2))$ for any $M$-category $\underline{B}$ with object set $B$.

Now suppose that $C$ is an $M$-category with object set $B$, that $D$ is an ordinary category with object set $A$, and $f:D \to C_0$ is a fully faithful and essentially surjective functor, where $C_0$ is the underlying ordinary category of $C$. Then $f$ induces a function $A\to B$ and thereby an induced $M$-category $f^*C$ with object set $A$. Moreover, the action of $f$ on homs consists of bijections $D(x,y) \cong C_0(f(x),f(y))$ that respect composition and identities; but by construction of $f^*C$ these are equivalently $D(x,y) \cong (f^*C)_0(x,y)$, giving an isomorphism of categories $D\cong (f^*C)_0$, i.e. an enrichment of $D$ over $M$.

Note that nowhere did we use the essential surjectivity of $f$, so actually we've proven a bit more.


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